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mw-first-heading"><span class="mw-page-title-main">Basis function</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 11 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-11" 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">11 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_base" title="Función base – Spanish" lang="es" hreflang="es" data-title="Función base" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Baza_funkcio" title="Baza funkcio – Esperanto" lang="eo" hreflang="eo" data-title="Baza funkcio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D8%A8%D9%86%DB%8C%D8%A7%D8%AF%DB%8C" title="تابع بنیادی – Persian" lang="fa" hreflang="fa" data-title="تابع بنیادی" 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/Fonction_de_base" title="Fonction de base – French" lang="fr" hreflang="fr" data-title="Fonction de base" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B8%B0%EC%A0%80_%ED%95%A8%EC%88%98" title="기저 함수 – Korean" lang="ko" hreflang="ko" data-title="기저 함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%AA_%D7%91%D7%A1%D7%99%D7%A1" title="פעולת בסיס – Hebrew" lang="he" hreflang="he" data-title="פעולת בסיס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9F%BA%E5%BA%95%E9%96%A2%E6%95%B0" title="基底関数 – Japanese" lang="ja" hreflang="ja" data-title="基底関数" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Basisfunksjon" title="Basisfunksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Basisfunksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B0%D0%B7%D0%B8%D1%81%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Базисная функция – Russian" lang="ru" hreflang="ru" data-title="Базисная функция" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D0%B0%D0%B7%D0%B8%D1%81%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Базисна функція – Ukrainian" lang="uk" hreflang="uk" data-title="Базисна функція" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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%9F%BA%E5%87%BD%E6%95%B8" title="基函數 – Chinese" lang="zh" hreflang="zh" data-title="基函數" data-language-autonym="中文" data-language-local-name="Chinese" 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/Q2621825#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</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="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > 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.compact-ambox .ambox .mbox-empty-cell,.mw-parser-output .compact-ambox .hide-when-compact{display:none}</style><table class="box-Multiple_issues plainlinks metadata ambox ambox-content ambox-multiple_issues compact-ambox" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span"><div class="multiple-issues-text mw-collapsible"><b>This article has multiple issues.</b> Please help <b><a href="/wiki/Special:EditPage/Basis_function" 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citations.</span> <span class="date-container"><i>(<span class="date">March 2013</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" 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metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Crystal_Clear_app_kedit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/40px-Crystal_Clear_app_kedit.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/60px-Crystal_Clear_app_kedit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/80px-Crystal_Clear_app_kedit.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may need to be rewritten</b> to comply with Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>.<span class="hide-when-compact"> <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Basis_function&action=edit">You can help</a>. The <a href="/wiki/Talk:Basis_function" title="Talk:Basis function">talk page</a> may contain suggestions.</span> <span class="date-container"><i>(<span class="date">September 2019</span>)</i></span></div></td></tr></tbody></table> </div> </div><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>basis function</b> is an element of a particular <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> for a <a href="/wiki/Function_space" title="Function space">function space</a>. Every <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> in the function space can be represented as a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of basis functions, just as every vector in a <a href="/wiki/Vector_space" title="Vector space">vector space</a> can be represented as a linear combination of <a href="/wiki/Basis_vectors" class="mw-redirect" title="Basis vectors">basis vectors</a>. </p><p>In <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a> and <a href="/wiki/Approximation_theory" title="Approximation theory">approximation theory</a>, basis functions are also called <b>blending functions,</b> because of their use in <a href="/wiki/Interpolation" title="Interpolation">interpolation</a>: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Basis_function&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Monomial_basis_for_Cω"><span id="Monomial_basis_for_C.CF.89"></span>Monomial basis for <i>C<sup>ω</sup></i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Basis_function&action=edit&section=2" title="Edit section: Monomial basis for Cω"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Monomial" title="Monomial">monomial</a> basis for the vector space of <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x^{n}\mid n\in \mathbb {N} \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∣</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x^{n}\mid n\in \mathbb {N} \}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1c75719738455b22e5ac9d23d0846849cd25010" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.371ex; height:2.843ex;" alt="{\displaystyle \{x^{n}\mid n\in \mathbb {N} \}.}"></span> </p><p>This basis is used in <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>, amongst others. </p> <div class="mw-heading mw-heading3"><h3 id="Monomial_basis_for_polynomials">Monomial basis for polynomials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Basis_function&action=edit&section=3" title="Edit section: Monomial basis for polynomials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The monomial basis also forms a basis for the vector space of <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>. After all, every polynomial can be written as <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}x^{1}+a_{2}x^{2}+\cdots +a_{n}x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}+a_{1}x^{1}+a_{2}x^{2}+\cdots +a_{n}x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3efffde2e354a6b0b141fa48055711df8361f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.701ex; height:3.009ex;" alt="{\displaystyle a_{0}+a_{1}x^{1}+a_{2}x^{2}+\cdots +a_{n}x^{n}}"></span> for some <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\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span>, which is a linear combination of monomials. </p> <div class="mw-heading mw-heading3"><h3 id="Fourier_basis_for_L2[0,1]"><span id="Fourier_basis_for_L2.5B0.2C1.5D"></span>Fourier basis for <i>L</i><sup>2</sup>[0,1]</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Basis_function&action=edit&section=4" title="Edit section: Fourier basis for L2[0,1]"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Trigonometric_functions" title="Trigonometric functions">Sines and cosines</a> form an (<a href="/wiki/Orthonormality" title="Orthonormality">orthonormal</a>) <a href="/wiki/Schauder_basis" title="Schauder basis">Schauder basis</a> for <a href="/wiki/Square-integrable_function" title="Square-integrable function">square-integrable functions</a> on a bounded domain. As a particular example, the collection <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\sqrt {2}}\sin(2\pi nx)\mid n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\mid n\in \mathbb {N} \}\cup \{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\sqrt {2}}\sin(2\pi nx)\mid n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\mid n\in \mathbb {N} \}\cup \{1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f06ed9d0b7096311cff7549b5e70c94893c0b47" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.997ex; height:3.176ex;" alt="{\displaystyle \{{\sqrt {2}}\sin(2\pi nx)\mid n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\mid n\in \mathbb {N} \}\cup \{1\}}"></span> forms a basis for <a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup>2</sup>[0,1]</a>. </p> <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=Basis_function&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis (linear algebra)</a> (<a href="/wiki/Hamel_basis" class="mw-redirect" title="Hamel basis">Hamel basis</a>)</li> <li><a href="/wiki/Schauder_basis" title="Schauder basis">Schauder basis</a> (in a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>)</li> <li><a href="/wiki/Dual_basis" title="Dual basis">Dual basis</a></li> <li><a href="/wiki/Biorthogonal_system" title="Biorthogonal system">Biorthogonal system</a> (Markushevich basis)</li> <li><a href="/wiki/Orthonormal_basis" title="Orthonormal basis">Orthonormal basis</a> in an <a href="/wiki/Inner-product_space" class="mw-redirect" title="Inner-product space">inner-product space</a></li> <li><a href="/wiki/Orthogonal_polynomials" title="Orthogonal polynomials">Orthogonal polynomials</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a> and <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Orthogonal_wavelet" title="Orthogonal wavelet">Orthogonal wavelet</a></li> <li><a href="/wiki/Biorthogonal_wavelet" title="Biorthogonal wavelet">Biorthogonal wavelet</a></li> <li><a href="/wiki/Radial_basis_function" title="Radial basis function">Radial basis function</a></li> <li><a href="/wiki/Finite_element_analysis#Choosing_a_basis" class="mw-redirect" title="Finite element analysis">Finite-elements (bases)</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Approximation_theory" title="Approximation theory">Approximation theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li></ul> </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=Basis_function&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFItô1993" class="citation book cs1">Itô, Kiyosi (1993). <i>Encyclopedic Dictionary of Mathematics</i> (2nd ed.). MIT Press. p. 1141. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-59020-4" title="Special:BookSources/0-262-59020-4"><bdi>0-262-59020-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedic+Dictionary+of+Mathematics&rft.pages=1141&rft.edition=2nd&rft.pub=MIT+Press&rft.date=1993&rft.isbn=0-262-59020-4&rft.aulast=It%C3%B4&rft.aufirst=Kiyosi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABasis+function" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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=Basis_function&oldid=1099683576">https://en.wikipedia.org/w/index.php?title=Basis_function&oldid=1099683576</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">Categories</a>: <ul><li><a href="/wiki/Category:Numerical_analysis" title="Category:Numerical analysis">Numerical analysis</a></li><li><a href="/wiki/Category:Fourier_analysis" title="Category:Fourier analysis">Fourier analysis</a></li><li><a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Linear algebra</a></li><li><a href="/wiki/Category:Numerical_linear_algebra" title="Category:Numerical linear algebra">Numerical linear algebra</a></li><li><a href="/wiki/Category:Types_of_functions" title="Category:Types of functions">Types of functions</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_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_March_2013" title="Category:Articles lacking in-text citations from March 2013">Articles lacking in-text citations from March 2013</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:Wikipedia_articles_that_are_too_technical_from_September_2019" title="Category:Wikipedia articles that are too technical from September 2019">Wikipedia articles that are too technical from September 2019</a></li><li><a href="/wiki/Category:All_articles_that_are_too_technical" title="Category:All articles that are too technical">All articles that are too technical</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_rewrite_from_September_2019" title="Category:Wikipedia articles needing rewrite from September 2019">Wikipedia articles needing rewrite from September 2019</a></li><li><a href="/wiki/Category:All_articles_needing_rewrite" title="Category:All articles needing rewrite">All articles needing rewrite</a></li><li><a href="/wiki/Category:Articles_with_multiple_maintenance_issues" title="Category:Articles with multiple maintenance issues">Articles with multiple maintenance issues</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 22 July 2022, at 02:35<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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