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((Short description|Element of a basis for a function space))
((Multiple issues|
((more footnotes|date=March 2013))
((Technical|date=September 2019))
((Cleanup rewrite|date=September 2019))
))
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 ''C<sup>ω</sup>''===
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''<sup>2</sup>[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''<sup>2</sup>[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]] <!-- shape functions in the [[Galerkin method]] and -->
* [[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]]

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