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Wavelet Haar - Wikipedia
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</a> </li> <li id="toc-Proprietà" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proprietà"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Proprietà</span> </div> </a> <ul id="toc-Proprietà-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrice_di_Haar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matrice_di_Haar"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Matrice di Haar</span> </div> </a> <ul id="toc-Matrice_di_Haar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </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="Indice" 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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Wavelet Haar</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="Vai a una voce in un'altra lingua. Disponibile in 15 lingue" > <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 lingue</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 - catalano" lang="ca" hreflang="ca" data-title="Ondeta de Haar" data-language-autonym="Català" data-language-local-name="catalano" 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 - ceco" lang="cs" hreflang="cs" data-title="Haarova vlnka" data-language-autonym="Čeština" data-language-local-name="ceco" 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 - tedesco" lang="de" hreflang="de" data-title="Haar-Wavelet" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Haar_wavelet" title="Haar wavelet - inglese" lang="en" hreflang="en" data-title="Haar wavelet" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</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 - spagnolo" lang="es" hreflang="es" data-title="Ondícula de Haar" data-language-autonym="Español" data-language-local-name="spagnolo" 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 - persiano" lang="fa" hreflang="fa" data-title="موجک هار" data-language-autonym="فارسی" data-language-local-name="persiano" 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 - francese" lang="fr" hreflang="fr" data-title="Ondelette de Haar" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</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 - giapponese" 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="giapponese" class="interlanguage-link-target"><span>日本語</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 - lituano" lang="lt" hreflang="lt" data-title="Haaro vilnelė" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Falki_Haara" title="Falki Haara - polacco" lang="pl" hreflang="pl" data-title="Falki Haara" data-language-autonym="Polski" data-language-local-name="polacco" 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 - portoghese" lang="pt" hreflang="pt" data-title="Transformada de Haar" data-language-autonym="Português" data-language-local-name="portoghese" 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 - russo" lang="ru" hreflang="ru" data-title="Вейвлет Хаара" data-language-autonym="Русский" data-language-local-name="russo" 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 - svedese" lang="sv" hreflang="sv" data-title="Haars wavelet" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B0%D0%B0%D1%80%D1%96%D0%B2_%D0%B2%D0%B5%D0%B9%D0%B2%D0%BB%D0%B5%D1%82" title="Гаарів вейвлет - ucraino" lang="uk" hreflang="uk" data-title="Гаарів вейвлет" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%93%88%E7%88%BE%E5%B0%8F%E6%B3%A2%E8%BD%89%E6%8F%9B" title="哈爾小波轉換 - cinese" lang="zh" hreflang="zh" data-title="哈爾小波轉換" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q766198#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespace"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Wavelet_Haar" title="Vedi la voce [c]" accesskey="c"><span>Voce</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussione:Wavelet_Haar" rel="discussion" title="Vedi le discussioni relative a questa pagina [t]" accesskey="t"><span>Discussione</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown 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class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Wavelet_Haar"><span>Leggi</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Wavelet_Haar&veaction=edit" title="Modifica questa pagina [v]" accesskey="v"><span>Modifica</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Wavelet_Haar&action=edit" title="Modifica il wikitesto di questa pagina [e]" accesskey="e"><span>Modifica wikitesto</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Wavelet_Haar&action=history" title="Versioni precedenti di questa pagina [h]" accesskey="h"><span>Cronologia</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" 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> <div class="vector-pinnable-header-label">Strumenti</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">nascondi</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Altre opzioni" > <div class="vector-menu-heading"> Azioni </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Wavelet_Haar"><span>Leggi</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Wavelet_Haar&veaction=edit" title="Modifica questa pagina [v]" accesskey="v"><span>Modifica</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Wavelet_Haar&action=edit" title="Modifica il wikitesto di questa pagina [e]" accesskey="e"><span>Modifica wikitesto</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Wavelet_Haar&action=history"><span>Cronologia</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Generale </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speciale:PuntanoQui/Wavelet_Haar" title="Elenco di tutte le pagine che sono collegate a questa [j]" accesskey="j"><span>Puntano qui</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speciale:ModificheCorrelate/Wavelet_Haar" rel="nofollow" title="Elenco delle ultime modifiche alle pagine collegate a questa [k]" accesskey="k"><span>Modifiche correlate</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speciale:PagineSpeciali" title="Elenco di tutte le pagine speciali [q]" accesskey="q"><span>Pagine speciali</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Wavelet_Haar&oldid=140364358" title="Collegamento permanente a questa versione di questa pagina"><span>Link permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Wavelet_Haar&action=info" title="Ulteriori informazioni su questa pagina"><span>Informazioni pagina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speciale:Cita&page=Wavelet_Haar&id=140364358&wpFormIdentifier=titleform" title="Informazioni su come citare questa pagina"><span>Cita questa voce</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speciale:UrlShortener&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FWavelet_Haar"><span>Ottieni URL breve</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speciale:QrCode&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FWavelet_Haar"><span>Scarica codice QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Stampa/esporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speciale:Libro&bookcmd=book_creator&referer=Wavelet+Haar"><span>Crea un libro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speciale:DownloadAsPdf&page=Wavelet_Haar&action=show-download-screen"><span>Scarica come PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Wavelet_Haar&printable=yes" title="Versione stampabile di questa pagina [p]" accesskey="p"><span>Versione stampabile</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In altri progetti </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Haar_wavelet" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q766198" title="Collegamento all'elemento connesso dell'archivio dati [g]" accesskey="g"><span>Elemento Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspetto"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">nascondi</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Da Wikipedia, l'enciclopedia libera.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><p>La <b>wavelet Haar</b> è stata la prima <a href="/wiki/Wavelet" title="Wavelet">wavelet</a> ad essere proposta nel <a href="/wiki/1909" title="1909">1909</a> da <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 usò queste funzioni per dare un esempio di un sistema ortonormale numerabile per lo spazio delle funzioni L2 sulla <a href="/wiki/Retta_reale" class="mw-redirect" title="Retta reale">retta reale</a>. </p><p>La wavelet Haar è anche la wavelet più semplice. Lo svantaggio della wavelet di Haar è che non è una <a href="/wiki/Funzione_continua" title="Funzione continua">funzione continua</a> e quindi non è <a href="/wiki/Derivata" title="Derivata">derivabile</a>. </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>La wavelet Haar</figcaption></figure> <p>La wavelet madre di Haar è la funzione </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<1/2,\\-1&1/2\leq t<1,\\0&{\mbox{altrimenti.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ<!-- 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> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</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>altrimenti.</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<1/2,\\-1&1/2\leq t<1,\\0&{\mbox{altrimenti.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1e8efc735e67a48ed33243b91c1b960ed696b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:28.219ex; height:8.509ex;" alt="{\displaystyle \psi (t)={\begin{cases}1\quad &0\leq t<1/2,\\-1&1/2\leq t<1,\\0&{\mbox{altrimenti.}}\end{cases}}}"></span></dd></dl> <p>e la sua funzione padre </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 \phi (t)={\begin{cases}1\quad &0\leq t<1,\\0&{\mbox{altrimenti.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ<!-- 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>altrimenti.</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (t)={\begin{cases}1\quad &0\leq t<1,\\0&{\mbox{altrimenti.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be2df382a868635686070faf39ea7728059b81d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.167ex; height:6.176ex;" alt="{\displaystyle \phi (t)={\begin{cases}1\quad &0\leq t<1,\\0&{\mbox{altrimenti.}}\end{cases}}}"></span></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Proprietà"><span id="Propriet.C3.A0"></span>Proprietà</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet_Haar&veaction=edit&section=1" title="Modifica la sezione Proprietà" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wavelet_Haar&action=edit&section=1" title="Edit section's source code: Proprietà"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La wavelet di Haar ha diverse proprietà: </p> <ul><li>Ogni funzione sufficientemente regolare può essere approssimata, in un senso che può essere precisato, da una <a href="/wiki/Combinazione_lineare" title="Combinazione lineare">combinazione lineare</a> di <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 \phi (t),\phi (2t),\phi (4t),\dots ,\phi (2^{k}t),\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…<!-- … --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (t),\phi (2t),\phi (4t),\dots ,\phi (2^{k}t),\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1071c1ff4cfec5a85fae6d7fc25441e7c79f2ca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.717ex; height:3.176ex;" alt="{\displaystyle \phi (t),\phi (2t),\phi (4t),\dots ,\phi (2^{k}t),\dots }"></span> e le loro traslazioni.</li> <li>Ogni funzione può essere approssimata dalla funzione costante 1 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 \psi (t),\psi (2t),\psi (4t),\dots ,\psi (2^{k}t),\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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^{k}t),\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5635d0e76dc2034642c13d99548ce325bcf797e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.228ex; height:3.176ex;" alt="{\displaystyle \psi (t),\psi (2t),\psi (4t),\dots ,\psi (2^{k}t),\dots }"></span> e le loro traslazioni.</li> <li><a href="/w/index.php?title=Ortonormalit%C3%A0&action=edit&redlink=1" class="new" title="Ortonormalità (la pagina non esiste)">Ortonormalità</a></li></ul> <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^{m}\psi (2^{m_{1}}t-n_{1})\psi (2^{m}t-n)\,dt=\delta (m-m_{1})\delta (n-n_{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"> <mi>m</mi> </mrow> </msup> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mi>t</mi> <mo>−<!-- − --></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mi>t</mi> <mo>−<!-- − --></mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>δ<!-- δ --></mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−<!-- − --></mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>δ<!-- δ --></mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }2^{m}\psi (2^{m_{1}}t-n_{1})\psi (2^{m}t-n)\,dt=\delta (m-m_{1})\delta (n-n_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f50eda6beb8897a7557dc2e0f141974bb03e7ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.943ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }2^{m}\psi (2^{m_{1}}t-n_{1})\psi (2^{m}t-n)\,dt=\delta (m-m_{1})\delta (n-n_{1})}"></span></dd></dl> <p>La <a href="/w/index.php?title=Funzione_duale&action=edit&redlink=1" class="new" title="Funzione duale (la pagina non esiste)">funzione duale</a> di <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<!-- 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> è <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<!-- 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> stessa. </p> <ul><li>Relazione madre/padre con diversa scala <i>m</i>:</li></ul> <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 \phi (t)=\phi (2t)+\phi (2t-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (t)=\phi (2t)+\phi (2t-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff15c73378cb7b09e8ec47451be0475236e439a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.37ex; height:2.843ex;" alt="{\displaystyle \phi (t)=\phi (2t)+\phi (2t-1)}"></span></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 \psi (t)=\phi (2t)-\phi (2t-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t)=\phi (2t)-\phi (2t-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bde9d228f291148b779d3f29500e229cbfdb012f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.497ex; height:2.843ex;" alt="{\displaystyle \psi (t)=\phi (2t)-\phi (2t-1)}"></span></dd></dl> <ul><li>I coefficienti di scala <i>m</i> possono essere calcolati dai coefficienti di scala <i>m+1</i></li></ul> <p>Se <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}(n,m)=2^{m/2}\int _{-\infty }^{\infty }x(t)\phi (2^{m}t-n)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ<!-- χ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</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<!-- ϕ --></mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mi>t</mi> <mo>−<!-- − --></mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{w}(n,m)=2^{m/2}\int _{-\infty }^{\infty }x(t)\phi (2^{m}t-n)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/387d94dbd52e8feba75963a49d7f9601bc43373d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.471ex; height:6.009ex;" alt="{\displaystyle \chi _{w}(n,m)=2^{m/2}\int _{-\infty }^{\infty }x(t)\phi (2^{m}t-n)\,dt}"></span> </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 \chi _{w}(n,m)={\sqrt {\frac {1}{2}}}(\chi _{w}(2n,m+1)+\chi _{w}(2n+1,m+1))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ<!-- χ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mi>χ<!-- χ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>χ<!-- χ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{w}(n,m)={\sqrt {\frac {1}{2}}}(\chi _{w}(2n,m+1)+\chi _{w}(2n+1,m+1))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97fdd63f6459f0138120647b2b3c2cde9a7fcf66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.831ex; height:6.176ex;" alt="{\displaystyle \chi _{w}(n,m)={\sqrt {\frac {1}{2}}}(\chi _{w}(2n,m+1)+\chi _{w}(2n+1,m+1))}"></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}(n,m)=2^{m/2}\int _{-\infty }^{\infty }x(t)\psi (2^{m}t-n)\,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>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</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<!-- ψ --></mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mi>t</mi> <mo>−<!-- − --></mo> <mi>n</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}(n,m)=2^{m/2}\int _{-\infty }^{\infty }x(t)\psi (2^{m}t-n)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e9ef684bd6535ffd0164142196d13ddfa038a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.887ex; height:6.009ex;" alt="{\displaystyle \mathrm {X} _{w}(n,m)=2^{m/2}\int _{-\infty }^{\infty }x(t)\psi (2^{m}t-n)\,dt}"></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}(n,m)={\sqrt {\frac {1}{2}}}(\chi _{w}(2n,m+1)-\chi _{w}(2n+1,m+1))}"> <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>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mi>χ<!-- χ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <msub> <mi>χ<!-- χ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {X} _{w}(n,m)={\sqrt {\frac {1}{2}}}(\chi _{w}(2n,m+1)-\chi _{w}(2n+1,m+1))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67f5c4c5483238aa568dce488188d009a87df94b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.119ex; height:6.176ex;" alt="{\displaystyle \mathrm {X} _{w}(n,m)={\sqrt {\frac {1}{2}}}(\chi _{w}(2n,m+1)-\chi _{w}(2n+1,m+1))}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Matrice_di_Haar">Matrice di Haar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet_Haar&veaction=edit&section=2" title="Modifica la sezione Matrice di Haar" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wavelet_Haar&action=edit&section=2" title="Edit section's source code: Matrice di Haar"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La matrice di Haar 2×2 associata con la wavelet è </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>Usando la <a href="/w/index.php?title=Trasformata_wavelet_discreta&action=edit&redlink=1" class="new" title="Trasformata wavelet discreta (la pagina non esiste)">trasformata wavelet discreta</a> si può trasformare ogni sequenza <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> di lunghezza pari in una sequenza di vettori a due componenti <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),\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> <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),\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/edb6c160e04d1b19e503fd4238708640aa6cba83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.27ex; height:2.843ex;" alt="{\displaystyle \left(\left(a_{0},a_{1}\right),\dots ,\left(a_{2n},a_{2n+1}\right)\right)}"></span>. Se si moltiplica ogni vettore con la matrice <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> si ottiene il risultato <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>, </p><p>Se si hanno sequenze di lunghezza multiplo di quattro si possono costruire blocchi di 4 elementi e trasformarli in maniera simile con una matrice di Haar 4×4 </p><p><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> </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/c83ff3e98b25d82c0d38bf32103b1729837b45d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:26.978ex; 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>, </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet_Haar&veaction=edit&section=3" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wavelet_Haar&action=edit&section=3" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation pubblicazione" style="font-style:normal"> Alfréd Haar, <a rel="nofollow" class="external text" href="http://gdz-srv1.sub.uni-goettingen.de:8080/contentserver/contentserver?pdf.x=23&pdf.y=7&pdf=pdf&command=docconvert&docid=D28629&pid=GDZPPN002263440"><span style="font-style:italic;">Zur Theorie der orthogonalen Funktionensysteme. (Erste Mitteilung).</span></a> <style data-mw-deduplicate="TemplateStyles:r140554517">.mw-parser-output .chiarimento{background:#ffeaea;color:#444444}.mw-parser-output .chiarimento-apice{color:#EE0700}@media screen{html.skin-theme-clientpref-night .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-night .mw-parser-output .chiarimento-apice{color:#b32424}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-os .mw-parser-output .chiarimento-apice{color:#b32424}}</style><span class="chiarimento" title="A volte può capitare che un link presente su Wikipedia non sia più raggiungibile. Se possibile ritrova il link e inserisci il collegamento corretto, comunque non rimuovere il collegamento e inserisci il template {{Collegamento interrotto}}"></span><sup class="noprint chiarimento-apice" title="A volte può capitare che un link presente su Wikipedia non sia più raggiungibile. Se possibile ritrova il link e inserisci il collegamento corretto, comunque non rimuovere il collegamento e inserisci il template {{Collegamento interrotto}}">[<i><a href="/wiki/Aiuto:Collegamenti_interrotti" title="Aiuto:Collegamenti interrotti">collegamento interrotto</a></i>]</sup>, in <span style="font-style:italic;">Mathematische Annalen</span>, vol. 69, n. 3, pp. 331-371, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2FBF01456326">10.1007/BF01456326</a>. <small>URL consultato il 29 settembre 2008</small>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet_Haar&veaction=edit&section=4" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wavelet_Haar&action=edit&section=4" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Charles K. Chui, <i>An Introduction to Wavelets</i>, (1992), Academic Press, San Diego, <a href="/wiki/Speciale:RicercaISBN/0585470901" class="internal mw-magiclink-isbn">ISBN 0585470901</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet_Haar&veaction=edit&section=5" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wavelet_Haar&action=edit&section=5" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Matrice_di_Walsh" title="Matrice di Walsh">Matrice di Walsh</a></li> <li><a href="/wiki/Segnale_(fisica)" class="mw-redirect" title="Segnale (fisica)">Segnale (fisica)</a></li> <li><a href="/wiki/Computer_vision" class="mw-redirect" title="Computer vision">Computer vision</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet_Haar&veaction=edit&section=6" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wavelet_Haar&action=edit&section=6" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><span class="plainlinks" title="commons:Category:Haar wavelet"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Haar_wavelet?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file su <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Haar_wavelet?uselang=it">wavelet Haar</a></span></b></li></ul> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Brain_art.png" class="mw-file-description" title="Neuroscienze"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Brain_art.png/25px-Brain_art.png" decoding="async" width="25" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Brain_art.png/38px-Brain_art.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Brain_art.png/50px-Brain_art.png 2x" data-file-width="100" data-file-height="92" /></a></span> <b><a href="/wiki/Portale:Neuroscienze" title="Portale:Neuroscienze">Portale Neuroscienze</a></b>: accedi alle voci di Wikipedia che trattano di neuroscienze</div></div></div> <!-- NewPP limit report Parsed by mw‐web.eqiad.main‐7c479b968‐dw8vq Cached time: 20241117172459 Cache expiry: 2592000 Reduced expiry: false Complications: [show‐toc] CPU time usage: 0.128 seconds Real time usage: 0.504 seconds Preprocessor visited node count: 1488/1000000 Post‐expand include size: 7193/2097152 bytes Template argument size: 197/2097152 bytes Highest expansion depth: 9/100 Expensive parser function count: 1/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 3972/5000000 bytes Lua time usage: 0.061/10.000 seconds Lua memory usage: 1878695/52428800 bytes Number of Wikibase entities loaded: 1/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 134.779 1 -total 33.78% 45.523 1 Template:Cita_pubblicazione 25.39% 34.218 1 Template:En 23.90% 32.217 1 Template:Lingue 21.04% 28.359 1 Template:Portale 18.41% 24.819 1 Template:Interprogetto 10.91% 14.706 1 Template:Chiarimento 10.54% 14.209 1 Template:Icona_argomento --> <!-- Saved in parser cache with key itwiki:pcache:idhash:1995038-0!canonical and timestamp 20241117172459 and revision id 140364358. 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