imported>OCR2718: Double hyperlink to "linear combination" in previous edit

2025-12-14T08:24:18Z

Double hyperlink to "linear combination" in previous edit

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	In [[mathematics]], a '''basis function''' is an element of a particular [[Basis (linear algebra)\|basis]] for a [[function space]]. Every ~~[[function (mathematics)\|~~function]] in the function space can be represented as a [[linear combination]] of basis functions, ~~just as every vector~~ in a [[~~vector space~~]] ~~can be represented as a linear combination of~~ [[~~basis vectors~~]].		In [[mathematics]], a '''basis function''' is an element of a particular [[Basis (linear algebra)\|basis]] for a [[function space]]. Every function in the function space can be represented as a [[linear combination]] of basis functions. In finite-dimensional [[vector spaces]] this representation is purely algebraic and involves only finitely many basis functions, whereas in infinite-dimensional settings it typically takes the form of an [[infinite series]] whose convergence depends on the [[topology]] of the space.

	In [[numerical analysis]] and [[approximation theory]], basis functions are also called '''blending functions,''' because of their use in [[interpolation]]: 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).		In [[numerical analysis]] and [[approximation theory]], basis functions are also called '''blending functions,''' because of their use in [[interpolation]]: 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).

	==Examples==		==Examples==

imported>OlliverWithDoubleL at 02:35, 22 July 2022

2022-07-22T02:35:05Z

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{{Short description|Element of a basis for a function space}}
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{{more footnotes|date=March 2013}}
{{Technical|date=September 2019}}
{{Cleanup rewrite|date=September 2019}}
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In [[mathematics]], a '''basis function''' is an element of a particular [[Basis (linear algebra)|basis]] for a [[function space]]. Every [[function (mathematics)|function]] in the function space can be represented as a [[linear combination]] of basis functions, just as every vector in a [[vector space]] can be represented as a linear combination of [[basis vectors]].

In [[numerical analysis]] and [[approximation theory]], basis functions are also called '''blending functions,''' because of their use in [[interpolation]]: 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).

==Examples==

===Monomial basis for ''Cω''===
The [[monomial]] basis for the vector space of [[analytic function]]s is given by
<math display="block">\{x^n \mid n\in\N\}.</math>

This basis is used in [[Taylor series]], amongst others.

===Monomial basis for polynomials===
The monomial basis also forms a basis for the vector space of [[polynomial]]s. After all, every polynomial can be written as <math>a_0 + a_1x^1 + a_2x^2 + \cdots + a_n x^n</math> for some <math>n \in \mathbb{N}</math>, which is a linear combination of monomials.

===Fourier basis for ''L''2[0,1]===
[[Trigonometric functions|Sines and cosines]] form an ([[orthonormality|orthonormal]]) [[Schauder basis]] for [[square-integrable function]]s on a bounded domain. As a particular example, the collection
<math display="block">\{\sqrt{2}\sin(2\pi n x) \mid n \in \N \} \cup \{\sqrt{2} \cos(2\pi n x) \mid n \in \N \} \cup \{1\}</math>
forms a basis for [[Lp space|''L''2[0,1]]].

==See also==
{{div col|colwidth=22em}}
* [[Basis (linear algebra)]] ([[Hamel basis]])
* [[Schauder basis]] (in a [[Banach space]])
* [[Dual basis]]
* [[Biorthogonal system]] (Markushevich basis)
* [[Orthonormal basis]] in an [[inner-product space]]
* [[Orthogonal polynomials]]
* [[Fourier analysis]] and [[Fourier series]]
* [[Harmonic analysis]]
* [[Orthogonal wavelet]]
* [[Biorthogonal wavelet]]
* [[Radial basis function]] 
* [[Finite element analysis#Choosing a basis|Finite-elements (bases)]]
* [[Functional analysis]]
* [[Approximation theory]]
* [[Numerical analysis]]
{{div col end}}

==References==
<references />
*{{cite book |last=Itô |first=Kiyosi |title=Encyclopedic Dictionary of Mathematics |edition=2nd |year=1993 |publisher=MIT Press |isbn=0-262-59020-4 | page=1141}}

[[Category:Numerical analysis]]
[[Category:Fourier analysis]]
[[Category:Linear algebra]]
[[Category:Numerical linear algebra]]
[[Category:Types of functions]]

Basis function - Revision history

imported>OCR2718: Double hyperlink to "linear combination" in previous edit

imported>OlliverWithDoubleL at 02:35, 22 July 2022