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<title>Basis function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Basis function</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>basis function</b> is an element of a particular <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> for a <a href="Function_space" title="Function space">function space</a>. Every <a href="Function_(mathematics)" title="Function (mathematics)">function</a> in the function space can be represented as a <a href="Linear_combination" title="Linear combination">linear combination</a> of basis functions, just as every vector in a <a href="Vector_space" title="Vector space">vector space</a> can be represented as a linear combination of <a href="Basis_vectors" class="mw-redirect" title="Basis vectors">basis vectors</a>.

</p><p>In <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a> and <a href="Approximation_theory" title="Approximation theory">approximation theory</a>, basis functions are also called <b>blending functions,</b> because of their use in <a href="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).

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Monomial_basis_for_Cω">Monomial basis for <i>C<sup>ω</sup></i></h3></div>

<p>The <a href="Monomial" title="Monomial">monomial</a> basis for the vector space of <a href="Analytic_function" title="Analytic function">analytic functions</a> is given by

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<annotation encoding="application/x-tex">{\displaystyle \{x^{n}\mid n\in \mathbb {N} \}.}</annotation>

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</p><p>This basis is used in <a href="Taylor_series" title="Taylor series">Taylor series</a>, amongst others.

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<div class="mw-heading mw-heading3"><h3 id="Monomial_basis_for_polynomials">Monomial basis for polynomials</h3></div>

<p>The monomial basis also forms a basis for the vector space of <a href="Polynomial" title="Polynomial">polynomials</a>. After all, every polynomial can be written as <span class="mwe-math-element mwe-math-element-inline"><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}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0}+a_{1}x^{1}+a_{2}x^{2}+\cdots +a_{n}x^{n}}</annotation>

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</math></span><img src="./7e3efffde2e354a6b0b141fa48055711df8361f7.svg" 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}}" loading="lazy"></span> for some <span class="mwe-math-element mwe-math-element-inline"><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} }">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

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</math></span><img src="./d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" 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} }" loading="lazy"></span>, which is a linear combination of monomials.

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<div class="mw-heading mw-heading3"><h3 id="Fourier_basis_for_L2[0,1]">Fourier basis for <i>L</i><sup>2</sup>[0,1]</h3></div>

<p><a href="Trigonometric_functions" title="Trigonometric functions">Sines and cosines</a> form an (<a href="Orthonormality" title="Orthonormality">orthonormal</a>) <a href="Schauder_basis" title="Schauder basis">Schauder basis</a> for <a href="Square-integrable_function" title="Square-integrable function">square-integrable functions</a> on a bounded domain. As a particular example, the collection

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<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>

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forms a basis for <a href="Lp_space" title="Lp space"><i>L</i><sup>2</sup>[0,1]</a>.

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<ul><li><a href="Basis_(linear_algebra)" title="Basis (linear algebra)">Basis (linear algebra)</a> (<a href="Hamel_basis" class="mw-redirect" title="Hamel basis">Hamel basis</a>)</li>

<li><a href="Schauder_basis" title="Schauder basis">Schauder basis</a> (in a <a href="Banach_space" title="Banach space">Banach space</a>)</li>

<li><a href="Dual_basis" title="Dual basis">Dual basis</a></li>

<li><a href="Biorthogonal_system" title="Biorthogonal system">Biorthogonal system</a> (Markushevich basis)</li>

<li><a href="Orthonormal_basis" title="Orthonormal basis">Orthonormal basis</a> in an <a href="Inner-product_space" class="mw-redirect" title="Inner-product space">inner-product space</a></li>

<li><a href="Orthogonal_polynomials" title="Orthogonal polynomials">Orthogonal polynomials</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a> and <a href="Fourier_series" title="Fourier series">Fourier series</a></li>

<li><a href="Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li>

<li><a href="Orthogonal_wavelet" title="Orthogonal wavelet">Orthogonal wavelet</a></li>

<li><a href="Biorthogonal_wavelet" title="Biorthogonal wavelet">Biorthogonal wavelet</a></li>

<li><a href="Radial_basis_function" title="Radial basis function">Radial basis function</a></li>

<li><a href="Finite_element_analysis" class="mw-redirect" title="Finite element analysis">Finite-elements (bases)</a></li>

<li><a href="Functional_analysis" title="Functional analysis">Functional analysis</a></li>

<li><a href="Approximation_theory" title="Approximation theory">Approximation theory</a></li>

<li><a href="Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</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="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-59020-4</bdi>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2022-07-22" href="https://en.wikipedia.org/wiki/?title=Basis_function&oldid=1099683576">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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