Haar wavelet - Wikipedia

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class="vector-toc-link" href="#Haar_wavelet_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Haar wavelet properties</span> </div> </a> <ul id="toc-Haar_wavelet_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Haar_system_on_the_unit_interval_and_related_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Haar_system_on_the_unit_interval_and_related_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Haar system on the unit interval and related systems</span> </div> </a> <button aria-controls="toc-Haar_system_on_the_unit_interval_and_related_systems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Haar system on the unit interval and related systems subsection</span> </button> <ul id="toc-Haar_system_on_the_unit_interval_and_related_systems-sublist" class="vector-toc-list"> <li id="toc-The_Faber–Schauder_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Faber–Schauder_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>The Faber–Schauder system</span> </div> </a> <ul id="toc-The_Faber–Schauder_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Franklin_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Franklin_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>The Franklin system</span> </div> </a> <ul id="toc-The_Franklin_system-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Haar_matrix" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Haar_matrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Haar matrix</span> </div> </a> <ul id="toc-Haar_matrix-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Haar_transform" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Haar_transform"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Haar transform</span> </div> </a> <button aria-controls="toc-Haar_transform-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Haar transform subsection</span> </button> <ul id="toc-Haar_transform-sublist" class="vector-toc-list"> <li id="toc-Introduction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Introduction"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Introduction</span> </div> </a> <ul id="toc-Introduction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Property" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Property"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Property</span> </div> </a> <ul id="toc-Property-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Haar_transform_and_Inverse_Haar_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Haar_transform_and_Inverse_Haar_transform"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Haar transform and Inverse Haar transform</span> </div> </a> <ul id="toc-Haar_transform_and_Inverse_Haar_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Haar_transform_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Haar_transform_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Haar transform</span> </div> </a> <ul id="toc-Haar_transform_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Haar wavelet</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 15 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-15" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">15 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Ondeta_de_Haar" title="Ondeta de Haar – Catalan" lang="ca" hreflang="ca" data-title="Ondeta de Haar" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Haarova_vlnka" title="Haarova vlnka – Czech" lang="cs" hreflang="cs" data-title="Haarova vlnka" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Haar-Wavelet" title="Haar-Wavelet – German" lang="de" hreflang="de" data-title="Haar-Wavelet" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ond%C3%ADcula_de_Haar" title="Ondícula de Haar – Spanish" lang="es" hreflang="es" data-title="Ondícula de Haar" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%88%D8%AC%DA%A9_%D9%87%D8%A7%D8%B1" title="موجک هار – Persian" lang="fa" hreflang="fa" data-title="موجک هار" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Ondelette_de_Haar" title="Ondelette de Haar – French" lang="fr" hreflang="fr" data-title="Ondelette de Haar" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Wavelet_Haar" title="Wavelet Haar – Italian" lang="it" hreflang="it" data-title="Wavelet Haar" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Haaro_vilnel%C4%97" title="Haaro vilnelė – Lithuanian" lang="lt" hreflang="lt" data-title="Haaro vilnelė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%8F%E3%83%BC%E3%83%AB%E3%82%A6%E3%82%A7%E3%83%BC%E3%83%96%E3%83%AC%E3%83%83%E3%83%88" title="ハールウェーブレット – Japanese" lang="ja" hreflang="ja" data-title="ハールウェーブレット" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Falki_Haara" title="Falki Haara – Polish" lang="pl" hreflang="pl" data-title="Falki Haara" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Transformada_de_Haar" title="Transformada de Haar – Portuguese" lang="pt" hreflang="pt" data-title="Transformada de Haar" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D0%B9%D0%B2%D0%BB%D0%B5%D1%82_%D0%A5%D0%B0%D0%B0%D1%80%D0%B0" title="Вейвлет Хаара – Russian" lang="ru" hreflang="ru" data-title="Вейвлет Хаара" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Haars_wavelet" title="Haars wavelet – Swedish" lang="sv" hreflang="sv" data-title="Haars wavelet" data-language-autonym="Svenska" data-language-local-name="Swedish" 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class="cdx-message__content"><div id="mw-revision-info-current"><p><b>This is the <a href="/wiki/Help:Page_history" title="Help:Page history">current revision</a> of this page, as edited by <span id="mw-revision-name"><a href="/wiki/User:Cewbot" class="mw-userlink" title="User:Cewbot" data-mw-revid="1237425461"><bdi>Cewbot</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Cewbot" class="mw-redirect mw-usertoollinks-talk" title="User talk:Cewbot">talk</a> | <a href="/wiki/Special:Contributions/Cewbot" class="mw-usertoollinks-contribs" title="Special:Contributions/Cewbot">contribs</a>)</span></span> at <span id="mw-revision-date">18:50, 29 July 2024</span><span id="mw-revision-summary"> <span class="comment">(<a href="/wiki/User:Cewbot/log/20201008/configuration" title="User:Cewbot/log/20201008/configuration">Fixing broken anchor</a>: Reminder of an inactive anchor: <a href="/wiki/Orthogonality#Orthogonal_functions" title="Orthogonality">orthogonal</a>)</span></span>. The present address (URL) is a <a href="/wiki/Help:Permanent_link" title="Help:Permanent link">permanent link</a> to this version.</b></p><div id="revision-info-current-plain" style="display: none;">Revision as of 18:50, 29 July 2024 by <a href="/wiki/User:Cewbot" class="mw-userlink" title="User:Cewbot" data-mw-revid="1237425461"><bdi>Cewbot</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Cewbot" class="mw-redirect mw-usertoollinks-talk" title="User talk:Cewbot">talk</a> | <a href="/wiki/Special:Contributions/Cewbot" class="mw-usertoollinks-contribs" title="Special:Contributions/Cewbot">contribs</a>)</span> <span class="comment">(<a href="/wiki/User:Cewbot/log/20201008/configuration" title="User:Cewbot/log/20201008/configuration">Fixing broken anchor</a>: Reminder of an inactive anchor: <a href="/wiki/Orthogonality#Orthogonal_functions" title="Orthogonality">orthogonal</a>)</span></div></div><div id="mw-revision-nav">(<a href="/w/index.php?title=Haar_wavelet&diff=prev&oldid=1237425461" title="Haar wavelet">diff</a>) <a href="/w/index.php?title=Haar_wavelet&direction=prev&oldid=1237425461" title="Haar wavelet">← Previous revision</a> | Latest revision (diff) | Newer revision → (diff)</div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">First known wavelet basis</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Haar_wavelet.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Haar_wavelet.svg/220px-Haar_wavelet.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Haar_wavelet.svg/330px-Haar_wavelet.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Haar_wavelet.svg/440px-Haar_wavelet.svg.png 2x" data-file-width="1300" data-file-height="975" /></a><figcaption>The Haar wavelet</figcaption></figure> <p>In mathematics, the <b>Haar wavelet</b> is a sequence of rescaled "square-shaped" functions which together form a <a href="/wiki/Wavelet" title="Wavelet">wavelet</a> family or basis. Wavelet analysis is similar to <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a> in that it allows a target function over an interval to be represented in terms of an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a>. The Haar sequence is now recognised as the first known wavelet basis and is extensively used as a teaching example. </p><p>The <b>Haar sequence</b> was proposed in 1909 by <a href="/wiki/Alfr%C3%A9d_Haar" title="Alfréd Haar">Alfréd Haar</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Haar used these functions to give an example of an orthonormal system for the space of <a href="/wiki/Square-integrable_function" title="Square-integrable function">square-integrable functions</a> on the <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a> [0, 1]. The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the <a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies wavelet</a>, the Haar wavelet is also known as <b>Db1</b>. </p><p>The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is not <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>, and therefore not <a href="/wiki/Derivative" title="Derivative">differentiable</a>. This property can, however, be an advantage for the analysis of signals with sudden transitions (<a href="/wiki/Digital_signal_(signal_processing)" title="Digital signal (signal processing)">discrete signals</a>), such as monitoring of tool failure in machines.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The Haar wavelet's mother wavelet function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cf4a36b5f945be90a527b3dbe3d55d3f0439cdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.162ex; height:2.843ex;" alt="{\displaystyle \psi (t)}"></span> can be described as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t)={\begin{cases}1\quad &0\leq t<{\frac {1}{2}},\\-1&{\frac {1}{2}}\leq t<1,\\0&{\mbox{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mspace width="1em" /> </mtd> <mtd> <mn>0</mn> <mo>≤</mo> <mi>t</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>≤</mo> <mi>t</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>otherwise.</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t)={\begin{cases}1\quad &0\leq t<{\frac {1}{2}},\\-1&{\frac {1}{2}}\leq t<1,\\0&{\mbox{otherwise.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49b8adc97251c218768de82f6d77d8f7ea45a6c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:26.39ex; height:9.843ex;" alt="{\displaystyle \psi (t)={\begin{cases}1\quad &0\leq t<{\frac {1}{2}},\\-1&{\frac {1}{2}}\leq t<1,\\0&{\mbox{otherwise.}}\end{cases}}}"></span></dd></dl> <p>Its <a href="/wiki/Father_wavelets" class="mw-redirect" title="Father wavelets">scaling function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd7776ed1f48fe8d39d9852e6ed6aa8a61a93d28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \varphi (t)}"></span> can be described as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)={\begin{cases}1\quad &0\leq t<1,\\0&{\mbox{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mspace width="1em" /> </mtd> <mtd> <mn>0</mn> <mo>≤</mo> <mi>t</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>otherwise.</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (t)={\begin{cases}1\quad &0\leq t<1,\\0&{\mbox{otherwise.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e6d3fa8b014ab9ead71dd9d85049270981d886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.794ex; height:6.176ex;" alt="{\displaystyle \varphi (t)={\begin{cases}1\quad &0\leq t<1,\\0&{\mbox{otherwise.}}\end{cases}}}"></span></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Haar_functions_and_Haar_system">Haar functions and Haar system</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=1" title="Edit section: Haar functions and Haar system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For every pair <i>n</i>, <i>k</i> of integers in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>, the <b>Haar function</b> <i>ψ</i><sub><i>n</i>,<i>k</i></sub> is defined on the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> by the formula </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{n,k}(t)=2^{n/2}\psi (2^{n}t-k),\quad t\in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>t</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n,k}(t)=2^{n/2}\psi (2^{n}t-k),\quad t\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fdf1a4bcadc77cebd2a1fb34192af7817e97dd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.774ex; height:3.509ex;" alt="{\displaystyle \psi _{n,k}(t)=2^{n/2}\psi (2^{n}t-k),\quad t\in \mathbb {R} .}"></span></dd></dl> <p>This function is supported on the <a href="/wiki/Semi-open_interval" class="mw-redirect" title="Semi-open interval">right-open interval</a> <span class="nowrap"> <i>I</i><sub><i>n</i>,<i>k</i></sub> =</span> <span class="nowrap">[ <i>k</i>2<sup>−<i>n</i></sup>, (<i>k</i>+1)2<sup>−<i>n</i></sup>)</span>, <i>i.e.</i>, it <a href="/wiki/Zero_of_a_function" title="Zero of a function">vanishes</a> outside that interval. It has integral 0 and norm 1 in the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> <a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup>2</sup>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>)</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} }\psi _{n,k}(t)\,dt=0,\quad \|\psi _{n,k}\|_{L^{2}(\mathbb {R} )}^{2}=\int _{\mathbb {R} }\psi _{n,k}(t)^{2}\,dt=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} }\psi _{n,k}(t)\,dt=0,\quad \|\psi _{n,k}\|_{L^{2}(\mathbb {R} )}^{2}=\int _{\mathbb {R} }\psi _{n,k}(t)^{2}\,dt=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/499013fe7b4afb3056204c1b8e540ef48820d9a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.168ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} }\psi _{n,k}(t)\,dt=0,\quad \|\psi _{n,k}\|_{L^{2}(\mathbb {R} )}^{2}=\int _{\mathbb {R} }\psi _{n,k}(t)^{2}\,dt=1.}"></span></dd></dl> <p>The Haar functions are pairwise <a href="/wiki/Orthogonality#Orthogonal_functions" title="Orthogonality">orthogonal</a><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/MOS:BROKENSECTIONLINKS" class="mw-redirect" title="MOS:BROKENSECTIONLINKS"><span title="The anchor (Orthogonal functions) has been deleted. (2024-07-29)">broken anchor</span></a></i>]</sup>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} }\psi _{n_{1},k_{1}}(t)\psi _{n_{2},k_{2}}(t)\,dt=\delta _{n_{1}n_{2}}\delta _{k_{1}k_{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} }\psi _{n_{1},k_{1}}(t)\psi _{n_{2},k_{2}}(t)\,dt=\delta _{n_{1}n_{2}}\delta _{k_{1}k_{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/707055d79bff7418bdf1fb91f7f1e2ec606a6a2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.542ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} }\psi _{n_{1},k_{1}}(t)\psi _{n_{2},k_{2}}(t)\,dt=\delta _{n_{1}n_{2}}\delta _{k_{1}k_{2}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa75d04c11480d976e1396951e02cbb3c4f71568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}"></span> represents the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>. Here is the reason for orthogonality: when the two supporting intervals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{n_{1},k_{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{n_{1},k_{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef8e376d84deb5e8ef6f7d4819d5dc18baedb2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.219ex; height:2.843ex;" alt="{\displaystyle I_{n_{1},k_{1}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{n_{2},k_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{n_{2},k_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/634500574a66a169a0571c03a4bb38ac3a735ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.219ex; height:2.843ex;" alt="{\displaystyle I_{n_{2},k_{2}}}"></span> are not equal, then they are either disjoint, or else the smaller of the two supports, say <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{n_{1},k_{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{n_{1},k_{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef8e376d84deb5e8ef6f7d4819d5dc18baedb2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.219ex; height:2.843ex;" alt="{\displaystyle I_{n_{1},k_{1}}}"></span>, is contained in the lower or in the upper half of the other interval, on which the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{n_{2},k_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n_{2},k_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a61b8632d891822bcf36aa34ffd16e50106f1f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.709ex; height:2.843ex;" alt="{\displaystyle \psi _{n_{2},k_{2}}}"></span> remains constant. It follows in this case that the product of these two Haar functions is a multiple of the first Haar function, hence the product has integral 0. </p><p>The <b>Haar system</b> on the real line is the set of functions </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\psi _{n,k}(t)\;:\;n\in \mathbb {Z} ,\;k\in \mathbb {Z} \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="thickmathspace" /> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\psi _{n,k}(t)\;:\;n\in \mathbb {Z} ,\;k\in \mathbb {Z} \}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d4bed8f1de08acd9c4f2ee2575f87fed684f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.961ex; height:3.009ex;" alt="{\displaystyle \{\psi _{n,k}(t)\;:\;n\in \mathbb {Z} ,\;k\in \mathbb {Z} \}.}"></span></dd></dl> <p>It is <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">complete</a> in <i>L</i><sup>2</sup>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>): <i>The Haar system on the line is an orthonormal basis in</i> <i>L</i><sup>2</sup>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>). </p> <div class="mw-heading mw-heading2"><h2 id="Haar_wavelet_properties">Haar wavelet properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=2" title="Edit section: Haar wavelet properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Haar wavelet has several notable properties: </p> <div><ol><li>Any continuous real function with compact support can be approximated uniformly by <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t),\varphi (2t),\varphi (4t),\dots ,\varphi (2^{n}t),\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (t),\varphi (2t),\varphi (4t),\dots ,\varphi (2^{n}t),\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a87f6f459c1777b766afe068cb1b2f72e38f3e89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.385ex; height:2.843ex;" alt="{\displaystyle \varphi (t),\varphi (2t),\varphi (4t),\dots ,\varphi (2^{n}t),\dots }"></span> and their shifted functions. This extends to those function spaces where any function therein can be approximated by continuous functions.</li><li>Any continuous real function on [0, 1] can be approximated uniformly on [0, 1] by linear combinations of the constant function <b>1</b>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t),\psi (2t),\psi (4t),\dots ,\psi (2^{n}t),\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t),\psi (2t),\psi (4t),\dots ,\psi (2^{n}t),\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fdeaed6c1d6ebc4be0160885aab4ad8c60fb15b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.357ex; height:2.843ex;" alt="{\displaystyle \psi (t),\psi (2t),\psi (4t),\dots ,\psi (2^{n}t),\dots }"></span> and their shifted functions.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li><li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a> in the form <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }2^{(n+n_{1})/2}\psi (2^{n}t-k)\psi (2^{n_{1}}t-k_{1})\,dt=\delta _{nn_{1}}\delta _{kk_{1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>t</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mi>t</mi> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }2^{(n+n_{1})/2}\psi (2^{n}t-k)\psi (2^{n_{1}}t-k_{1})\,dt=\delta _{nn_{1}}\delta _{kk_{1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21e73d7dd2c3f085d1612f9f55fc50790cc37ce6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.375ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }2^{(n+n_{1})/2}\psi (2^{n}t-k)\psi (2^{n_{1}}t-k_{1})\,dt=\delta _{nn_{1}}\delta _{kk_{1}}.}"></span></dd></dl> Here, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa75d04c11480d976e1396951e02cbb3c4f71568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}"></span> represents the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>. The <a href="/wiki/Dual_function" class="mw-redirect" title="Dual function">dual function</a> of ψ(<i>t</i>) is ψ(<i>t</i>) itself.</li><li>Wavelet/scaling functions with different scale <i>n</i> have a functional relationship:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> since <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varphi (t)&=\varphi (2t)+\varphi (2t-1)\\[.2em]\psi (t)&=\varphi (2t)-\varphi (2t-1),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi (t)&=\varphi (2t)+\varphi (2t-1)\\[.2em]\psi (t)&=\varphi (2t)-\varphi (2t-1),\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d131f2b222a5a5cdbbd698b7d1d3ee7ab42b723e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.172ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\varphi (t)&=\varphi (2t)+\varphi (2t-1)\\[.2em]\psi (t)&=\varphi (2t)-\varphi (2t-1),\end{aligned}}}"></span></dd></dl> it follows that coefficients of scale <i>n</i> can be calculated by coefficients of scale <i>n+1</i>:<br /> If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{w}(k,n)=2^{n/2}\int _{-\infty }^{\infty }x(t)\varphi (2^{n}t-k)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>t</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{w}(k,n)=2^{n/2}\int _{-\infty }^{\infty }x(t)\varphi (2^{n}t-k)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19f3298b927d30f364ae04021779659066783b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.68ex; height:6.009ex;" alt="{\displaystyle \chi _{w}(k,n)=2^{n/2}\int _{-\infty }^{\infty }x(t)\varphi (2^{n}t-k)\,dt}"></span><br /> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {X} _{w}(k,n)=2^{n/2}\int _{-\infty }^{\infty }x(t)\psi (2^{n}t-k)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>t</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {X} _{w}(k,n)=2^{n/2}\int _{-\infty }^{\infty }x(t)\psi (2^{n}t-k)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e6e733e2f5cf1e05fcf482c5784a3d79e8b12b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.961ex; height:6.009ex;" alt="{\displaystyle \mathrm {X} _{w}(k,n)=2^{n/2}\int _{-\infty }^{\infty }x(t)\psi (2^{n}t-k)\,dt}"></span><br /> then <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{w}(k,n)=2^{-1/2}{\bigl (}\chi _{w}(2k,n+1)+\chi _{w}(2k+1,n+1){\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{w}(k,n)=2^{-1/2}{\bigl (}\chi _{w}(2k,n+1)+\chi _{w}(2k+1,n+1){\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa6368291bb1d9ea68aae14a64f4d0890096518" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:52.481ex; height:3.509ex;" alt="{\displaystyle \chi _{w}(k,n)=2^{-1/2}{\bigl (}\chi _{w}(2k,n+1)+\chi _{w}(2k+1,n+1){\bigr )}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {X} _{w}(k,n)=2^{-1/2}{\bigl (}\chi _{w}(2k,n+1)-\chi _{w}(2k+1,n+1){\bigr )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {X} _{w}(k,n)=2^{-1/2}{\bigl (}\chi _{w}(2k,n+1)-\chi _{w}(2k+1,n+1){\bigr )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c2243c752550055ede98af8f9614dd23b6429ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.416ex; height:3.509ex;" alt="{\displaystyle \mathrm {X} _{w}(k,n)=2^{-1/2}{\bigl (}\chi _{w}(2k,n+1)-\chi _{w}(2k+1,n+1){\bigr )}.}"></span></dd></dl></li></ol></div> <div class="mw-heading mw-heading2"><h2 id="Haar_system_on_the_unit_interval_and_related_systems">Haar system on the unit interval and related systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=3" title="Edit section: Haar system on the unit interval and related systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this section, the discussion is restricted to the <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a> [0, 1] and to the Haar functions that are supported on [0, 1]. The system of functions considered by Haar in 1910,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> called the <b>Haar system on [0, 1]</b> in this article, consists of the subset of Haar wavelets defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{t\in [0,1]\mapsto \psi _{n,k}(t)\;:\;n,k\in \mathbb {N} \cup \{0\},\;0\leq k<2^{n}\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">↦</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="thickmathspace" /> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mn>0</mn> <mo>≤</mo> <mi>k</mi> <mo><</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{t\in [0,1]\mapsto \psi _{n,k}(t)\;:\;n,k\in \mathbb {N} \cup \{0\},\;0\leq k<2^{n}\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/871c1f99ec17b483f6dbe3463c04cf3c556a9735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.7ex; height:3.009ex;" alt="{\displaystyle \{t\in [0,1]\mapsto \psi _{n,k}(t)\;:\;n,k\in \mathbb {N} \cup \{0\},\;0\leq k<2^{n}\},}"></span></dd></dl> <p>with the addition of the constant function <b>1</b> on [0, 1]. </p><p>In <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> terms, this Haar system on [0, 1] is a complete orthonormal system, <i>i.e.</i>, an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a>, for the space <i>L</i><sup>2</sup>([0, 1]) of square integrable functions on the unit interval. </p><p>The Haar system on [0, 1] —with the constant function <b>1</b> as first element, followed with the Haar functions ordered according to the <a href="/wiki/Lexicographical_order" class="mw-redirect" title="Lexicographical order">lexicographic</a> ordering of couples <span class="nowrap">(<i>n</i>, <i>k</i>)</span>— is further a <a href="/wiki/Schauder_basis#Properties" title="Schauder basis">monotone</a> <a href="/wiki/Schauder_basis" title="Schauder basis">Schauder basis</a> for the space <a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup><i>p</i></sup>([0, 1])</a> when <span class="nowrap">1 ≤ <i>p</i> < ∞</span>.<sup id="cite_ref-L._Tzafriri,_1977_6-0" class="reference"><a href="#cite_note-L._Tzafriri,_1977-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This basis is <a href="/wiki/Schauder_basis#Unconditionality" title="Schauder basis">unconditional</a> when <span class="nowrap">1 < <i>p</i> < ∞</span>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>There is a related <a href="/wiki/Rademacher_system" title="Rademacher system">Rademacher system</a> consisting of sums of Haar functions, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}(t)=2^{-n/2}\sum _{k=0}^{2^{n}-1}\psi _{n,k}(t),\quad t\in [0,1],\ n\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mtext> </mtext> <mi>n</mi> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}(t)=2^{-n/2}\sum _{k=0}^{2^{n}-1}\psi _{n,k}(t),\quad t\in [0,1],\ n\geq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aee69350f41f40964c92dd012cb92e20bea0875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.28ex; height:7.509ex;" alt="{\displaystyle r_{n}(t)=2^{-n/2}\sum _{k=0}^{2^{n}-1}\psi _{n,k}(t),\quad t\in [0,1],\ n\geq 0.}"></span></dd></dl> <p>Notice that |<i>r</i><sub><i>n</i></sub>(<i>t</i>)| = 1 on [0, 1). This is an orthonormal system but it is not complete.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In the language of <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, the Rademacher sequence is an instance of a sequence of <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independent</a> <a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a> <a href="/wiki/Random_variables" class="mw-redirect" title="Random variables">random variables</a> with <a href="/wiki/Mean" title="Mean">mean</a> 0. The <a href="/wiki/Khintchine_inequality" title="Khintchine inequality">Khintchine inequality</a> expresses the fact that in all the spaces <i>L</i><sup><i>p</i></sup>([0, 1]), <span class="nowrap">1 ≤ <i>p</i> < ∞</span>, the Rademacher sequence is <a href="/wiki/Schauder_basis#Definitions" title="Schauder basis">equivalent</a> to the unit vector basis in ℓ<sup><i>2</i></sup>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In particular, the <a href="/wiki/Linear_span#Closed_linear_span" title="Linear span">closed linear span</a> of the Rademacher sequence in <i>L</i><sup><i>p</i></sup>([0, 1]), <span class="nowrap">1 ≤ <i>p</i> < ∞</span>, is <a href="/wiki/Isomorphic_normed_spaces" class="mw-redirect" title="Isomorphic normed spaces">isomorphic</a> to ℓ<sup><i>2</i></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="The_Faber–Schauder_system"><span id="The_Faber.E2.80.93Schauder_system"></span>The Faber–Schauder system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=4" title="Edit section: The Faber–Schauder system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Faber–Schauder system</b><sup id="cite_ref-Faber_11-0" class="reference"><a href="#cite_note-Faber-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> is the family of continuous functions on [0, 1] consisting of the constant function <b>1</b>, and of multiples of <a href="/wiki/Antiderivative" title="Antiderivative">indefinite integrals</a> of the functions in the Haar system on [0, 1], chosen to have norm 1 in the <a href="/wiki/Uniform_norm" title="Uniform norm">maximum norm</a>. This system begins with <i>s</i><sub>0</sub> = <b>1</b>, then <span class="nowrap"> <i>s</i><sub>1</sub>(<i>t</i>) = <i>t</i></span> is the indefinite integral vanishing at 0 of the function <b>1</b>, first element of the Haar system on [0, 1]. Next, for every integer <span class="nowrap"><i>n</i> ≥ 0</span>, functions <span class="nowrap"> <i>s</i><sub><i>n</i>,<i>k</i></sub></span> are defined by the formula </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n,k}(t)=2^{1+n/2}\int _{0}^{t}\psi _{n,k}(u)\,du,\quad t\in [0,1],\ 0\leq k<2^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo>≤</mo> <mi>k</mi> <mo><</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n,k}(t)=2^{1+n/2}\int _{0}^{t}\psi _{n,k}(u)\,du,\quad t\in [0,1],\ 0\leq k<2^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90c0a74f453073761b92204339bdd4dac475d8ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.579ex; height:6.176ex;" alt="{\displaystyle s_{n,k}(t)=2^{1+n/2}\int _{0}^{t}\psi _{n,k}(u)\,du,\quad t\in [0,1],\ 0\leq k<2^{n}.}"></span></dd></dl> <p>These functions <span class="nowrap"> <i>s</i><sub><i>n</i>,<i>k</i></sub></span> are continuous, <a href="/wiki/Piecewise_linear_function" title="Piecewise linear function">piecewise linear</a>, supported by the interval <span class="nowrap"> <i>I</i><sub><i>n</i>,<i>k</i></sub></span> that also supports <span class="nowrap"> ψ<sub><i>n</i>,<i>k</i></sub></span>. The function <span class="nowrap"> <i>s</i><sub><i>n</i>,<i>k</i></sub></span> is equal to 1 at the midpoint <span class="nowrap"> <i>x</i><sub><i>n</i>,<i>k</i></sub></span> of the interval <span class="nowrap"> <i>I</i><sub><i>n</i>,<i>k</i></sub></span>, linear on both halves of that interval. It takes values between 0 and 1 everywhere. </p><p>The Faber–Schauder system is a <a href="/wiki/Schauder_basis" title="Schauder basis">Schauder basis</a> for the space <i>C</i>([0, 1]) of continuous functions on [0, 1].<sup id="cite_ref-L._Tzafriri,_1977_6-1" class="reference"><a href="#cite_note-L._Tzafriri,_1977-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> For every <i>f</i> in <i>C</i>([0, 1]), the partial sum </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n+1}=a_{0}s_{0}+a_{1}s_{1}+\sum _{m=0}^{n-1}{\Bigl (}\sum _{k=0}^{2^{m}-1}a_{m,k}s_{m,k}{\Bigr )}\in C([0,1])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n+1}=a_{0}s_{0}+a_{1}s_{1}+\sum _{m=0}^{n-1}{\Bigl (}\sum _{k=0}^{2^{m}-1}a_{m,k}s_{m,k}{\Bigr )}\in C([0,1])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e9e0e8deb8e25308bca25d19b751e32606b8de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.2ex; height:7.509ex;" alt="{\displaystyle f_{n+1}=a_{0}s_{0}+a_{1}s_{1}+\sum _{m=0}^{n-1}{\Bigl (}\sum _{k=0}^{2^{m}-1}a_{m,k}s_{m,k}{\Bigr )}\in C([0,1])}"></span></dd></dl> <p>of the <a href="/wiki/Series_expansion" title="Series expansion">series expansion</a> of <i>f</i> in the Faber–Schauder system is the continuous piecewise linear function that agrees with <i>f</i> at the <span class="nowrap">2<sup><i>n</i></sup> + 1</span> points <span class="nowrap"><i>k</i>2<sup>−<i>n</i></sup></span>, where <span class="nowrap"> 0 ≤ <i>k</i> ≤ 2<sup><i>n</i></sup></span>. Next, the formula </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n+2}-f_{n+1}=\sum _{k=0}^{2^{n}-1}{\bigl (}f(x_{n,k})-f_{n+1}(x_{n,k}){\bigr )}s_{n,k}=\sum _{k=0}^{2^{n}-1}a_{n,k}s_{n,k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n+2}-f_{n+1}=\sum _{k=0}^{2^{n}-1}{\bigl (}f(x_{n,k})-f_{n+1}(x_{n,k}){\bigr )}s_{n,k}=\sum _{k=0}^{2^{n}-1}a_{n,k}s_{n,k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2afa28d59e9308c81f89bfab174945f42bf6a8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.56ex; height:7.509ex;" alt="{\displaystyle f_{n+2}-f_{n+1}=\sum _{k=0}^{2^{n}-1}{\bigl (}f(x_{n,k})-f_{n+1}(x_{n,k}){\bigr )}s_{n,k}=\sum _{k=0}^{2^{n}-1}a_{n,k}s_{n,k}}"></span></dd></dl> <p>gives a way to compute the expansion of <i>f</i> step by step. Since <i>f</i> is <a href="/wiki/Heine%E2%80%93Borel_theorem" title="Heine–Borel theorem">uniformly continuous</a>, the sequence {<i>f</i><sub><i>n</i></sub>} converges uniformly to <i>f</i>. It follows that the Faber–Schauder series expansion of <i>f</i> converges in <i>C</i>([0, 1]), and the sum of this series is equal to <i>f</i>. </p> <div class="mw-heading mw-heading3"><h3 id="The_Franklin_system">The Franklin system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=5" title="Edit section: The Franklin system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Franklin system</b> is obtained from the Faber–Schauder system by the <a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt orthonormalization procedure</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Since the Franklin system has the same <a href="/wiki/Linear_span" title="Linear span">linear span</a> as that of the Faber–Schauder system, this span is dense in <i>C</i>([0, 1]), hence in <i>L</i><sup>2</sup>([0, 1]). The Franklin system is therefore an orthonormal basis for <i>L</i><sup>2</sup>([0, 1]), consisting of continuous piecewise linear functions. P. Franklin proved in 1928 that this system is a Schauder basis for <i>C</i>([0, 1]).<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The Franklin system is also an unconditional Schauder basis for the space <i>L</i><sup><i>p</i></sup>([0, 1]) when <span class="nowrap">1 < <i>p</i> < ∞</span>.<sup id="cite_ref-Bo_17-0" class="reference"><a href="#cite_note-Bo-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The Franklin system provides a Schauder basis in the <a href="/wiki/Disk_algebra" title="Disk algebra">disk algebra</a> <i>A</i>(<i>D</i>).<sup id="cite_ref-Bo_17-1" class="reference"><a href="#cite_note-Bo-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> This was proved in 1974 by Bočkarev, after the existence of a basis for the disk algebra had remained open for more than forty years.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Bočkarev's construction of a Schauder basis in <i>A</i>(<i>D</i>) goes as follows: let <i>f</i> be a complex valued <a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz function</a> on [0, π]; then <i>f</i> is the sum of a <a href="/wiki/Fourier_series" title="Fourier series">cosine series</a> with <a href="/wiki/Absolute_convergence" title="Absolute convergence">absolutely summable</a> coefficients. Let <i>T</i>(<i>f</i>) be the element of <i>A</i>(<i>D</i>) defined by the complex <a href="/wiki/Power_series" title="Power series">power series</a> with the same coefficients, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{f:x\in [0,\pi ]\rightarrow \sum _{n=0}^{\infty }a_{n}\cos(nx)\right\}\longrightarrow \left\{T(f):z\rightarrow \sum _{n=0}^{\infty }a_{n}z^{n},\quad |z|\leq 1\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mi>f</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo stretchy="false">⟶</mo> <mrow> <mo>{</mo> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>z</mi> <mo stretchy="false">→</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{f:x\in [0,\pi ]\rightarrow \sum _{n=0}^{\infty }a_{n}\cos(nx)\right\}\longrightarrow \left\{T(f):z\rightarrow \sum _{n=0}^{\infty }a_{n}z^{n},\quad |z|\leq 1\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a695350de4ab02ef21d8223dba8e6d47dc5d3ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:73.529ex; height:7.509ex;" alt="{\displaystyle \left\{f:x\in [0,\pi ]\rightarrow \sum _{n=0}^{\infty }a_{n}\cos(nx)\right\}\longrightarrow \left\{T(f):z\rightarrow \sum _{n=0}^{\infty }a_{n}z^{n},\quad |z|\leq 1\right\}.}"></span></dd></dl> <p>Bočkarev's basis for <i>A</i>(<i>D</i>) is formed by the images under <i>T</i> of the functions in the Franklin system on [0, π]. Bočkarev's equivalent description for the mapping <i>T</i> starts by extending <i>f</i> to an <a href="/wiki/Even_and_odd_functions" title="Even and odd functions">even</a> Lipschitz function <i>g</i><sub>1</sub> on [−π, π], identified with a Lipschitz function on the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> <b>T</b>. Next, let <i>g</i><sub>2</sub> be the <a href="/wiki/Hardy_space_conjugate_function" class="mw-redirect" title="Hardy space conjugate function">conjugate function</a> of <i>g</i><sub>1</sub>, and define <i>T</i>(<i>f</i>) to be the function in <i>A</i>(<i>D</i>) whose value on the boundary <b>T</b> of <i>D</i> is equal to <span class="nowrap"><i>g</i><sub>1</sub> + i<i>g</i><sub>2</sub></span>. </p><p>When dealing with 1-periodic continuous functions, or rather with continuous functions <i>f</i> on [0, 1] such that <span class="nowrap"><i>f</i>(0) = <i>f</i>(1)</span>, one removes the function <span class="nowrap"> <i>s</i><sub>1</sub>(<i>t</i>) = <i>t</i></span> from the Faber–Schauder system, in order to obtain the <b>periodic Faber–Schauder system</b>. The <b>periodic Franklin system</b> is obtained by orthonormalization from the periodic Faber–-Schauder system.<sup id="cite_ref-Prz_19-0" class="reference"><a href="#cite_note-Prz-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> One can prove Bočkarev's result on <i>A</i>(<i>D</i>) by proving that the periodic Franklin system on [0, 2π] is a basis for a Banach space <i>A</i><sub><i>r</i></sub> isomorphic to <i>A</i>(<i>D</i>).<sup id="cite_ref-Prz_19-1" class="reference"><a href="#cite_note-Prz-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The space <i>A</i><sub><i>r</i></sub> consists of complex continuous functions on the unit circle <b>T</b> whose <a href="/wiki/Harmonic_conjugate" title="Harmonic conjugate">conjugate function</a> is also continuous. </p> <div class="mw-heading mw-heading2"><h2 id="Haar_matrix">Haar matrix</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=6" title="Edit section: Haar matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The 2×2 Haar matrix that is associated with the Haar wavelet is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{2}={\begin{bmatrix}1&1\\1&-1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{2}={\begin{bmatrix}1&1\\1&-1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d083ecfc4a57fe9ae4999e24bb38f693f0f520f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.393ex; height:6.176ex;" alt="{\displaystyle H_{2}={\begin{bmatrix}1&1\\1&-1\end{bmatrix}}.}"></span></dd></dl> <p>Using the <a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">discrete wavelet transform</a>, one can transform any sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{0},a_{1},\dots ,a_{2n},a_{2n+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{0},a_{1},\dots ,a_{2n},a_{2n+1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97aac13f6e5749c6a44a3d26a71f86f2b5d9d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.265ex; height:2.843ex;" alt="{\displaystyle (a_{0},a_{1},\dots ,a_{2n},a_{2n+1})}"></span> of even length into a sequence of two-component-vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\left(a_{0},a_{1}\right),\left(a_{2},a_{3}\right),\dots ,\left(a_{2n},a_{2n+1}\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\left(a_{0},a_{1}\right),\left(a_{2},a_{3}\right),\dots ,\left(a_{2n},a_{2n+1}\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0def17de3b3494c06df8ea32d57e5fac709b2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.103ex; height:2.843ex;" alt="{\displaystyle \left(\left(a_{0},a_{1}\right),\left(a_{2},a_{3}\right),\dots ,\left(a_{2n},a_{2n+1}\right)\right)}"></span>. If one right-multiplies each vector with the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}"></span>, one gets the result <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\left(s_{0},d_{0}\right),\dots ,\left(s_{n},d_{n}\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\left(s_{0},d_{0}\right),\dots ,\left(s_{n},d_{n}\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b08ee6b05f90f1c10dbeefc91e18846aa8a806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.205ex; height:2.843ex;" alt="{\displaystyle \left(\left(s_{0},d_{0}\right),\dots ,\left(s_{n},d_{n}\right)\right)}"></span> of one stage of the fast Haar-wavelet transform. Usually one separates the sequences <i>s</i> and <i>d</i> and continues with transforming the sequence <i>s</i>. Sequence <i>s</i> is often referred to as the <i>averages</i> part, whereas <i>d</i> is known as the <i>details</i> part.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>If one has a sequence of length a multiple of four, one can build blocks of 4 elements and transform them in a similar manner with the 4×4 Haar matrix </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{4}={\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\1&-1&0&0\\0&0&1&-1\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{4}={\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\1&-1&0&0\\0&0&1&-1\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61d520ebac49d8cf577dda83f38ed4faa03c860a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:27.625ex; height:12.509ex;" alt="{\displaystyle H_{4}={\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\1&-1&0&0\\0&0&1&-1\end{bmatrix}},}"></span></dd></dl> <p>which combines two stages of the fast Haar-wavelet transform. </p><p>Compare with a <a href="/wiki/Walsh_matrix" title="Walsh matrix">Walsh matrix</a>, which is a non-localized 1/–1 matrix. </p><p>Generally, the 2N×2N Haar matrix can be derived by the following equation. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{2N}={\begin{bmatrix}H_{N}\otimes [1,1]\\I_{N}\otimes [1,-1]\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>⊗</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>⊗</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{2N}={\begin{bmatrix}H_{N}\otimes [1,1]\\I_{N}\otimes [1,-1]\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ee3ceaac9913e277c7fbbfdeec3aa8ef55cfb01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.765ex; height:6.176ex;" alt="{\displaystyle H_{2N}={\begin{bmatrix}H_{N}\otimes [1,1]\\I_{N}\otimes [1,-1]\end{bmatrix}}}"></span></dd> <dd>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{N}={\begin{bmatrix}1&0&\dots &0\\0&1&\dots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\dots &1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{N}={\begin{bmatrix}1&0&\dots &0\\0&1&\dots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\dots &1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/067db96a2813c45128413e562faa29fe0d3d4d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:23.099ex; height:14.176ex;" alt="{\displaystyle I_{N}={\begin{bmatrix}1&0&\dots &0\\0&1&\dots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\dots &1\end{bmatrix}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \otimes }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }"></span> is the <a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a>.</dd></dl> <p>The <a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\otimes B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊗</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\otimes B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887ed3ce13b460df337cda66497515383ded3e5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\otimes B}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is an m×n matrix and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> is a p×q matrix, is expressed as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\otimes B={\begin{bmatrix}a_{11}B&\dots &a_{1n}B\\\vdots &\ddots &\vdots \\a_{m1}B&\dots &a_{mn}B\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊗</mo> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mi>B</mi> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mi>B</mi> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mi>B</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\otimes B={\begin{bmatrix}a_{11}B&\dots &a_{1n}B\\\vdots &\ddots &\vdots \\a_{m1}B&\dots &a_{mn}B\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835b4799553ab456e94527d2f9d0cf5653b0df1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:32.715ex; height:11.009ex;" alt="{\displaystyle A\otimes B={\begin{bmatrix}a_{11}B&\dots &a_{1n}B\\\vdots &\ddots &\vdots \\a_{m1}B&\dots &a_{mn}B\end{bmatrix}}.}"></span></dd></dl> <p>An un-normalized 8-point Haar matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc2a5f50ec7e313a6b6882a8e853dc7b3a1f9bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{8}}"></span> is shown below </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{8}={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&1&1&1&-1&-1&-1&-1\\1&1&-1&-1&0&0&0&0&\\0&0&0&0&1&1&-1&-1\\1&-1&0&0&0&0&0&0&\\0&0&1&-1&0&0&0&0\\0&0&0&0&1&-1&0&0&\\0&0&0&0&0&0&1&-1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{8}={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&1&1&1&-1&-1&-1&-1\\1&1&-1&-1&0&0&0&0&\\0&0&0&0&1&1&-1&-1\\1&-1&0&0&0&0&0&0&\\0&0&1&-1&0&0&0&0\\0&0&0&0&1&-1&0&0&\\0&0&0&0&0&0&1&-1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aad656f999725cdbd2c18b750520a3df9b0c9cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:51.12ex; height:25.509ex;" alt="{\displaystyle H_{8}={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&1&1&1&-1&-1&-1&-1\\1&1&-1&-1&0&0&0&0&\\0&0&0&0&1&1&-1&-1\\1&-1&0&0&0&0&0&0&\\0&0&1&-1&0&0&0&0\\0&0&0&0&1&-1&0&0&\\0&0&0&0&0&0&1&-1\end{bmatrix}}.}"></span></dd></dl> <p>Note that, the above matrix is an un-normalized Haar matrix. The Haar matrix required by the Haar transform should be normalized. </p><p>From the definition of the Haar matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, one can observe that, unlike the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> has only real elements (i.e., 1, -1 or 0) and is non-symmetric. </p><p>Take the 8-point Haar matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc2a5f50ec7e313a6b6882a8e853dc7b3a1f9bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{8}}"></span> as an example. The first row of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc2a5f50ec7e313a6b6882a8e853dc7b3a1f9bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{8}}"></span> measures the average value, and the second row of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc2a5f50ec7e313a6b6882a8e853dc7b3a1f9bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{8}}"></span> measures a low frequency component of the input vector. The next two rows are sensitive to the first and second half of the input vector respectively, which corresponds to moderate frequency components. The remaining four rows are sensitive to the four section of the input vector, which corresponds to high frequency components.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Haar_transform">Haar transform</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=7" title="Edit section: Haar transform"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Haar transform</b> is the simplest of the <a href="/wiki/Wavelet_transform" title="Wavelet transform">wavelet transforms</a>. This transform cross-multiplies a function against the Haar wavelet with various shifts and stretches, like the Fourier transform cross-multiplies a function against a sine wave with two phases and many stretches.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (June 2018)">clarification needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Introduction">Introduction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=8" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Haar transform is one of the oldest transform functions, proposed in 1910 by the Hungarian mathematician <a href="/wiki/Alfr%C3%A9d_Haar" title="Alfréd Haar">Alfréd Haar</a>. It is found effective in applications such as signal and image compression in electrical and computer engineering as it provides a simple and computationally efficient approach for analysing the local aspects of a signal. </p><p>The Haar transform is derived from the Haar matrix. An example of a 4×4 Haar transformation matrix is shown below. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\{\sqrt {2}}&-{\sqrt {2}}&0&0\\0&0&{\sqrt {2}}&-{\sqrt {2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\{\sqrt {2}}&-{\sqrt {2}}&0&0\\0&0&{\sqrt {2}}&-{\sqrt {2}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82b7c1bcb5c1619d42774034920ee700cb3362c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:34.912ex; height:13.176ex;" alt="{\displaystyle H_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\{\sqrt {2}}&-{\sqrt {2}}&0&0\\0&0&{\sqrt {2}}&-{\sqrt {2}}\end{bmatrix}}}"></span></dd></dl> <p>The Haar transform can be thought of as a sampling process in which rows of the transformation matrix act as samples of finer and finer resolution. </p><p>Compare with the <a href="/wiki/Walsh_transform" class="mw-redirect" title="Walsh transform">Walsh transform</a>, which is also 1/–1, but is non-localized. </p> <div class="mw-heading mw-heading3"><h3 id="Property">Property</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=9" title="Edit section: Property"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Haar transform has the following properties </p> <ol><li>No need for multiplications. It requires only additions and there are many elements with zero value in the Haar matrix, so the computation time is short. It is faster than <a href="/wiki/Walsh_transform" class="mw-redirect" title="Walsh transform">Walsh transform</a>, whose matrix is composed of +1 and −1.</li> <li>Input and output length are the same. However, the length should be a power of 2, i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=2^{k},k\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=2^{k},k\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/350836557579750ee10f25cddfe79cec97bb980e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.177ex; height:3.009ex;" alt="{\displaystyle N=2^{k},k\in \mathbb {N} }"></span>.</li> <li>It can be used to analyse the localized feature of signals. Due to the <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> property of the Haar function, the frequency components of input signal can be analyzed.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Haar_transform_and_Inverse_Haar_transform">Haar transform and Inverse Haar transform</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=10" title="Edit section: Haar transform and Inverse Haar transform"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Haar transform <i>y</i><sub><i>n</i></sub> of an n-input function <i>x</i><sub><i>n</i></sub> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n}=H_{n}x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{n}=H_{n}x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/923a1cb9a740b1866e5f271e559c258424a603d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.154ex; height:2.509ex;" alt="{\displaystyle y_{n}=H_{n}x_{n}}"></span></dd></dl> <p>The Haar transform matrix is real and orthogonal. Thus, the inverse Haar transform can be derived by the following equations. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=H^{*},H^{-1}=H^{T},{\text{ i.e. }}HH^{T}=I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> i.e. </mtext> </mrow> <mi>H</mi> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=H^{*},H^{-1}=H^{T},{\text{ i.e. }}HH^{T}=I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/387f46addc39a72775a04bb592e756d468476e3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.376ex; height:3.009ex;" alt="{\displaystyle H=H^{*},H^{-1}=H^{T},{\text{ i.e. }}HH^{T}=I}"></span></dd></dl> <dl><dd>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> is the identity matrix. For example, when n = 4</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{4}^{T}H_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&{\sqrt {2}}&0\\1&1&-{\sqrt {2}}&0\\1&-1&0&{\sqrt {2}}\\1&-1&0&-{\sqrt {2}}\end{bmatrix}}\cdot \;{\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\{\sqrt {2}}&-{\sqrt {2}}&0&0\\0&0&{\sqrt {2}}&-{\sqrt {2}}\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{4}^{T}H_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&{\sqrt {2}}&0\\1&1&-{\sqrt {2}}&0\\1&-1&0&{\sqrt {2}}\\1&-1&0&-{\sqrt {2}}\end{bmatrix}}\cdot \;{\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\{\sqrt {2}}&-{\sqrt {2}}&0&0\\0&0&{\sqrt {2}}&-{\sqrt {2}}\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966411692e7a9fdca2c51fb20979a753cfaa507e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:86.062ex; height:13.509ex;" alt="{\displaystyle H_{4}^{T}H_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&{\sqrt {2}}&0\\1&1&-{\sqrt {2}}&0\\1&-1&0&{\sqrt {2}}\\1&-1&0&-{\sqrt {2}}\end{bmatrix}}\cdot \;{\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\{\sqrt {2}}&-{\sqrt {2}}&0&0\\0&0&{\sqrt {2}}&-{\sqrt {2}}\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}}"></span></dd></dl> <p>Thus, the inverse Haar transform is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}=H^{T}y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}=H^{T}y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b87f8adf2d0edd391ac0f6be6ffbae54052c74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.497ex; height:3.009ex;" alt="{\displaystyle x_{n}=H^{T}y_{n}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=11" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Haar transform coefficients of a n=4-point signal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{4}=[1,2,3,4]^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{4}=[1,2,3,4]^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3508b509bd4ee4219b9238ba8a389c8de4321aa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.917ex; height:3.176ex;" alt="{\displaystyle x_{4}=[1,2,3,4]^{T}}"></span> can be found as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{4}=H_{4}x_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\{\sqrt {2}}&-{\sqrt {2}}&0&0\\0&0&{\sqrt {2}}&-{\sqrt {2}}\end{bmatrix}}{\begin{bmatrix}1\\2\\3\\4\end{bmatrix}}={\begin{bmatrix}5\\-2\\-1/{\sqrt {2}}\\-1/{\sqrt {2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{4}=H_{4}x_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\{\sqrt {2}}&-{\sqrt {2}}&0&0\\0&0&{\sqrt {2}}&-{\sqrt {2}}\end{bmatrix}}{\begin{bmatrix}1\\2\\3\\4\end{bmatrix}}={\begin{bmatrix}5\\-2\\-1/{\sqrt {2}}\\-1/{\sqrt {2}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed2a1aed3c955f579acf0523374a057c639b91b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:61.784ex; height:13.176ex;" alt="{\displaystyle y_{4}=H_{4}x_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\{\sqrt {2}}&-{\sqrt {2}}&0&0\\0&0&{\sqrt {2}}&-{\sqrt {2}}\end{bmatrix}}{\begin{bmatrix}1\\2\\3\\4\end{bmatrix}}={\begin{bmatrix}5\\-2\\-1/{\sqrt {2}}\\-1/{\sqrt {2}}\end{bmatrix}}}"></span></dd></dl> <p>The input signal can then be perfectly reconstructed by the inverse Haar transform </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x_{4}}}=H_{4}^{T}y_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&{\sqrt {2}}&0\\1&1&-{\sqrt {2}}&0\\1&-1&0&{\sqrt {2}}\\1&-1&0&-{\sqrt {2}}\end{bmatrix}}{\begin{bmatrix}5\\-2\\-1/{\sqrt {2}}\\-1/{\sqrt {2}}\end{bmatrix}}={\begin{bmatrix}1\\2\\3\\4\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x_{4}}}=H_{4}^{T}y_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&{\sqrt {2}}&0\\1&1&-{\sqrt {2}}&0\\1&-1&0&{\sqrt {2}}\\1&-1&0&-{\sqrt {2}}\end{bmatrix}}{\begin{bmatrix}5\\-2\\-1/{\sqrt {2}}\\-1/{\sqrt {2}}\end{bmatrix}}={\begin{bmatrix}1\\2\\3\\4\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bcf709bbd79875d8636a94f16a023b797f4cf89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:60.228ex; height:13.509ex;" alt="{\displaystyle {\hat {x_{4}}}=H_{4}^{T}y_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&{\sqrt {2}}&0\\1&1&-{\sqrt {2}}&0\\1&-1&0&{\sqrt {2}}\\1&-1&0&-{\sqrt {2}}\end{bmatrix}}{\begin{bmatrix}5\\-2\\-1/{\sqrt {2}}\\-1/{\sqrt {2}}\end{bmatrix}}={\begin{bmatrix}1\\2\\3\\4\end{bmatrix}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Dimension_reduction" class="mw-redirect" title="Dimension reduction">Dimension reduction</a></li> <li><a href="/wiki/Walsh_matrix" title="Walsh matrix">Walsh matrix</a></li> <li><a href="/wiki/Walsh_transform" class="mw-redirect" title="Walsh transform">Walsh transform</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Chirplet" class="mw-redirect" title="Chirplet">Chirplet</a></li> <li><a href="/wiki/Signal_(electrical_engineering)" class="mw-redirect" title="Signal (electrical engineering)">Signal</a></li> <li><a href="/wiki/Haar-like_feature" title="Haar-like feature">Haar-like feature</a></li> <li><a href="/wiki/Str%C3%B6mberg_wavelet" title="Strömberg wavelet">Strömberg wavelet</a></li> <li><a href="/wiki/Dyadic_transformation" title="Dyadic transformation">Dyadic transformation</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=13" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">see p. 361 in <a href="#CITEREFHaar1910">Haar (1910)</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLeeTarng1999" class="citation journal cs1">Lee, B.; Tarng, Y. S. (1999). "Application of the discrete wavelet transform to the monitoring of tool failure in end milling using the spindle motor current". <i>International Journal of Advanced Manufacturing Technology</i>. <b>15</b> (4): 238–243. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs001700050062">10.1007/s001700050062</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:109908427">109908427</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Advanced+Manufacturing+Technology&rft.atitle=Application+of+the+discrete+wavelet+transform+to+the+monitoring+of+tool+failure+in+end+milling+using+the+spindle+motor+current&rft.volume=15&rft.issue=4&rft.pages=238-243&rft.date=1999&rft_id=info%3Adoi%2F10.1007%2Fs001700050062&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A109908427%23id-name%3DS2CID&rft.aulast=Lee&rft.aufirst=B.&rft.au=Tarng%2C+Y.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">As opposed to the preceding statement, this fact is not obvious: see p. 363 in <a href="#CITEREFHaar1910">Haar (1910)</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVidakovic2010" class="citation book cs1">Vidakovic, Brani (2010). <i>Statistical Modeling by Wavelets</i>. Wiley Series in Probability and Statistics (2 ed.). pp. 60, 63. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9780470317020">10.1002/9780470317020</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780470317020" title="Special:BookSources/9780470317020"><bdi>9780470317020</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Modeling+by+Wavelets&rft.series=Wiley+Series+in+Probability+and+Statistics&rft.pages=60%2C+63&rft.edition=2&rft.date=2010&rft_id=info%3Adoi%2F10.1002%2F9780470317020&rft.isbn=9780470317020&rft.aulast=Vidakovic&rft.aufirst=Brani&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">p. 361 in <a href="#CITEREFHaar1910">Haar (1910)</a></span> </li> <li id="cite_note-L._Tzafriri,_1977-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-L._Tzafriri,_1977_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-L._Tzafriri,_1977_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">see p. 3 in <a href="/wiki/Joram_Lindenstrauss" title="Joram Lindenstrauss">J. Lindenstrauss</a>, L. Tzafriri, (1977), "Classical Banach Spaces I, Sequence Spaces", Ergebnisse der Mathematik und ihrer Grenzgebiete <b>92</b>, Berlin: Springer-Verlag, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-08072-4" title="Special:BookSources/3-540-08072-4">3-540-08072-4</a>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">The result is due to <a href="/wiki/Raymond_Paley" title="Raymond Paley">R. E. Paley</a>, <i>A remarkable series of orthogonal functions (I)</i>, Proc. London Math. Soc. <b>34</b> (1931) pp. 241-264. See also p. 155 in J. Lindenstrauss, L. Tzafriri, (1979), "Classical Banach spaces II, Function spaces". Ergebnisse der Mathematik und ihrer Grenzgebiete <b>97</b>, Berlin: Springer-Verlag, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-08888-1" title="Special:BookSources/3-540-08888-1">3-540-08888-1</a>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Orthogonal_system">"Orthogonal system"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Orthogonal+system&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DOrthogonal_system&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWalterShen2001" class="citation book cs1">Walter, Gilbert G.; Shen, Xiaoping (2001). <i>Wavelets and Other Orthogonal Systems</i>. Boca Raton: Chapman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-58488-227-1" title="Special:BookSources/1-58488-227-1"><bdi>1-58488-227-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wavelets+and+Other+Orthogonal+Systems&rft.place=Boca+Raton&rft.pub=Chapman&rft.date=2001&rft.isbn=1-58488-227-1&rft.aulast=Walter&rft.aufirst=Gilbert+G.&rft.au=Shen%2C+Xiaoping&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">see for example p. 66 in <a href="/wiki/Joram_Lindenstrauss" title="Joram Lindenstrauss">J. Lindenstrauss</a>, L. Tzafriri, (1977), "Classical Banach Spaces I, Sequence Spaces", Ergebnisse der Mathematik und ihrer Grenzgebiete <b>92</b>, Berlin: Springer-Verlag, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-08072-4" title="Special:BookSources/3-540-08072-4">3-540-08072-4</a>.</span> </li> <li id="cite_note-Faber-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Faber_11-0">^</a></b></span> <span class="reference-text">Faber, Georg (1910), "Über die Orthogonalfunktionen des Herrn Haar", <i>Deutsche Math.-Ver</i> (in German) <b>19</b>: 104–112. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/search?fq=x0:jrnl&q=n2:0012-0456">0012-0456</a>; <a rel="nofollow" class="external free" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X">http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X</a> ; <a rel="nofollow" class="external free" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002122553">http://resolver.sub.uni-goettingen.de/purl?GDZPPN002122553</a></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Schauder, Juliusz (1928), "Eine Eigenschaft des Haarschen Orthogonalsystems", <i>Mathematische Zeitschrift</i> <b>28</b>: 317–320.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubov2001" class="citation cs2">Golubov, B.I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Faber–Schauder_system">"Faber–Schauder system"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Faber%E2%80%93Schauder+system&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Golubov&rft.aufirst=B.I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DFaber%E2%80%93Schauder_system&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">see Z. Ciesielski, <i>Properties of the orthonormal Franklin system</i>. Studia Math. 23 1963 141–157.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Franklin system. B.I. Golubov (originator), Encyclopedia of Mathematics. URL: <a rel="nofollow" class="external free" href="http://www.encyclopediaofmath.org/index.php?title=Franklin_system&oldid=16655">http://www.encyclopediaofmath.org/index.php?title=Franklin_system&oldid=16655</a></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Philip Franklin, <i>A set of continuous orthogonal functions</i>, Math. Ann. 100 (1928), 522-529. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01448860">10.1007/BF01448860</a></span> </li> <li id="cite_note-Bo-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bo_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bo_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">S. V. Bočkarev, <i>Existence of a basis in the space of functions analytic in the disc, and some properties of Franklin's system</i>. Mat. Sb. <b>95</b> (1974), 3–18 (Russian). Translated in Math. USSR-Sb. <b>24</b> (1974), 1–16.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">The question appears p. 238, §3 in Banach's book, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBanach1932" class="citation cs2"><a href="/wiki/Stefan_Banach" title="Stefan Banach">Banach, Stefan</a> (1932), <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/kstresc.php?tom=1&wyd=10"><i>Théorie des opérations linéaires</i></a>, Monografie Matematyczne, vol. 1, Warszawa: Subwencji Funduszu Kultury Narodowej, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0005.20901">0005.20901</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+op%C3%A9rations+lin%C3%A9aires&rft.place=Warszawa&rft.series=Monografie+Matematyczne&rft.pub=Subwencji+Funduszu+Kultury+Narodowej&rft.date=1932&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0005.20901%23id-name%3DZbl&rft.aulast=Banach&rft.aufirst=Stefan&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fkstresc.php%3Ftom%3D1%26wyd%3D10&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span>. The disk algebra <i>A</i>(<i>D</i>) appears as Example 10, p. 12 in Banach's book.</span> </li> <li id="cite_note-Prz-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-Prz_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Prz_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">See p. 161, III.D.20 and p. 192, III.E.17 in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWojtaszczyk1991" class="citation cs2">Wojtaszczyk, Przemysław (1991), <i>Banach spaces for analysts</i>, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge: Cambridge University Press, pp. xiv+382, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-35618-0" title="Special:BookSources/0-521-35618-0"><bdi>0-521-35618-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Banach+spaces+for+analysts&rft.place=Cambridge&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pages=xiv%2B382&rft.pub=Cambridge+University+Press&rft.date=1991&rft.isbn=0-521-35618-0&rft.aulast=Wojtaszczyk&rft.aufirst=Przemys%C5%82aw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRuchVan_Fleet2009" class="citation book cs1">Ruch, David K.; Van Fleet, Patrick J. (2009). <i>Wavelet Theory: An Elementary Approach with Applications</i>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-38840-2" title="Special:BookSources/978-0-470-38840-2"><bdi>978-0-470-38840-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wavelet+Theory%3A+An+Elementary+Approach+with+Applications&rft.pub=John+Wiley+%26+Sons&rft.date=2009&rft.isbn=978-0-470-38840-2&rft.aulast=Ruch&rft.aufirst=David+K.&rft.au=Van+Fleet%2C+Patrick+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120821004423/http://fourier.eng.hmc.edu/e161/lectures/Haar/index.html">"haar"</a>. Fourier.eng.hmc.edu. 30 October 2013. Archived from <a rel="nofollow" class="external text" href="http://fourier.eng.hmc.edu/e161/lectures/Haar/index.html">the original</a> on 21 August 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">23 November</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=haar&rft.pub=Fourier.eng.hmc.edu&rft.date=2013-10-30&rft_id=http%3A%2F%2Ffourier.eng.hmc.edu%2Fe161%2Flectures%2FHaar%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://sepwww.stanford.edu/public/docs/sep75/ray2/paper_html/node4.html">The Haar Transform</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaar1910" class="citation cs2"><a href="/wiki/Alfr%C3%A9d_Haar" title="Alfréd Haar">Haar, Alfréd</a> (1910), "Zur Theorie der orthogonalen Funktionensysteme", <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>, <b>69</b> (3): 331–371, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01456326">10.1007/BF01456326</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fuc1.b2619563">2027/uc1.b2619563</a></span>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120024038">120024038</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Zur+Theorie+der+orthogonalen+Funktionensysteme&rft.volume=69&rft.issue=3&rft.pages=331-371&rft.date=1910&rft_id=info%3Ahdl%2F2027%2Fuc1.b2619563&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120024038%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01456326&rft.aulast=Haar&rft.aufirst=Alfr%C3%A9d&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></li> <li>Charles K. Chui, <i>An Introduction to Wavelets</i>, (1992), Academic Press, San Diego, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-585-47090-1" title="Special:BookSources/0-585-47090-1">0-585-47090-1</a></li> <li>English Translation of Haar's seminal article: <a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20120123124430/https://www.uni-hohenheim.de/~gzim/Publications/haar.pdf">[1]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Haar_wavelet" class="extiw" title="commons:Category:Haar wavelet">Haar wavelet</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Haar_system">"Haar system"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Haar+system&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DHaar_system&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.tomgibara.com/computer-vision/haar-wavelet">Free Haar wavelet filtering implementation and interactive demo</a></li> <li><a rel="nofollow" class="external text" href="http://packages.debian.org/wzip">Free Haar wavelet denoising and lossy signal compression</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Haar_transform_2">Haar transform</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Haar_wavelet&action=edit&section=16" title="Edit section: Haar transform"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKingsbury" class="citation web cs1">Kingsbury, Nick. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060419183057/http://cnx.org/content/m11087/latest/">"The Haar Transform"</a>. Archived from <a rel="nofollow" class="external text" href="http://cnx.org/content/m11087/latest/">the original</a> on 19 April 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Haar+Transform&rft.aulast=Kingsbury&rft.aufirst=Nick&rft_id=http%3A%2F%2Fcnx.org%2Fcontent%2Fm11087%2Flatest%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEck2006" class="citation web cs1">Eck, David (31 January 2006). <a rel="nofollow" class="external text" href="http://math.hws.edu/eck/math371/applets/Haar.html">"Haar Transform Demo Applets"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Haar+Transform+Demo+Applets&rft.date=2006-01-31&rft.aulast=Eck&rft.aufirst=David&rft_id=http%3A%2F%2Fmath.hws.edu%2Feck%2Fmath371%2Fapplets%2FHaar.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmes2002" class="citation web cs1">Ames, Greg (7 December 2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110125080404/http://online.redwoods.cc.ca.us/instruct/darnold/laproj/Fall2002/ames/paper.pdf">"Image Compression"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://online.redwoods.cc.ca.us/instruct/darnold/laproj/Fall2002/ames/paper.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 25 January 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Image+Compression&rft.date=2002-12-07&rft.aulast=Ames&rft.aufirst=Greg&rft_id=http%3A%2F%2Fonline.redwoods.cc.ca.us%2Finstruct%2Fdarnold%2Flaproj%2FFall2002%2Fames%2Fpaper.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAaronHillSrivatsa" class="citation web cs1">Aaron, Anne; Hill, Michael; Srivatsa, Anand. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080318071618/http://scien.stanford.edu/class/ee368/projects2000/project12/2.html">"MOSMAT 500. A photomosaic generator. 2. Theory"</a>. Archived from <a rel="nofollow" class="external text" href="http://scien.stanford.edu/class/ee368/projects2000/project12/2.html">the original</a> on 18 March 2008.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=MOSMAT+500.+A+photomosaic+generator.+2.+Theory&rft.aulast=Aaron&rft.aufirst=Anne&rft.au=Hill%2C+Michael&rft.au=Srivatsa%2C+Anand&rft_id=http%3A%2F%2Fscien.stanford.edu%2Fclass%2Fee368%2Fprojects2000%2Fproject12%2F2.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWang2008" class="citation web cs1">Wang, Ruye (4 December 2008). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120821004423/http://fourier.eng.hmc.edu/e161/lectures/Haar/index.html">"Haar Transform"</a>. Archived from <a rel="nofollow" class="external text" href="http://fourier.eng.hmc.edu/e161/lectures/Haar/index.html">the original</a> on 21 August 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Haar+Transform&rft.date=2008-12-04&rft.aulast=Wang&rft.aufirst=Ruye&rft_id=http%3A%2F%2Ffourier.eng.hmc.edu%2Fe161%2Flectures%2FHaar%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHaar+wavelet" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Haar_wavelet&oldid=1237425461">https://en.wikipedia.org/w/index.php?title=Haar_wavelet&oldid=1237425461</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Orthogonal_wavelets" title="Category:Orthogonal wavelets">Orthogonal wavelets</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_March_2024" title="Category:Use dmy dates from March 2024">Use dmy dates from March 2024</a></li><li><a href="/wiki/Category:Pages_with_broken_anchors" title="Category:Pages with broken anchors">Pages with broken anchors</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_June_2018" title="Category:Wikipedia articles needing clarification from June 2018">Wikipedia articles needing clarification from June 2018</a></li><li><a href="/wiki/Category:Commons_category_link_from_Wikidata" title="Category:Commons category link from Wikidata">Commons category link from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 29 July 2024, at 18:50<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Haar wavelet - Wikipedia