Basis function
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/* Fourier basis */ removing second link to Schauder basis; eliminating pipe on first one
In [[mathematics]], particularly [[numerical analysis]], a '''basis function''' is an element of the [[Basis (linear algebra)|basis]] for a [[function space]]. The term is a degeneration of the term ''basis vector'' for a more general [[vector space]]; that is, each function in the function space can be represented as a [[linear combination]] of the basis functions.
== Examples ==
===Polynomial bases===
The collection of quadratic polynomials with real coefficients has {1, ''t'', ''t''<sup>2</sup>} as a basis. Every quadratic can be written as ''a''1+''bt''+''ct''<sup>2</sup>, that is, as a [[linear combination]] of the basis functions 1, ''t'', and ''t''<sup>2</sup>. The set {(1/2)(''t''-1)(''t''-2), -''t''(''t''-2), (1/2)''t''(''t''-1)} is another basis for quadratic polynomials, called the [[Lagrange polynomial|Lagrange basis]].
===Fourier basis===
Sines and cosines form an ([[orthonormality|orthonormal]]) [[Schauder basis]] for square-integrable functions. As a particular example, the collection:
:<math>\{\sin(n\pi x) \; | \; n\in\mathbb{Z} \; \text{and} \; n\geq 1\} \cup \{\cos(n\pi x) \; | \; n\in\mathbb{Z} \; \text{and} \; n\geq 0\}</math>
forms a basis for [[Lebesgue space|L<sup>2</sup>(0,1)]].
==References==
*{{cite book |last=Ito |first=Kiyosi |authorlink= |coauthors= |others= |title=Encyclopedic Dictionary of Mathematics |edition=2nd ed. |year=1993 |publisher=MIT Press |location= |id=ISBN 0262590204 | pages=p. 1141}}
== See also ==
* [[Orthogonal polynomials]]
* [[Radial basis function]]
* shape functions in the [[Galerkin method]] and [[finite element analysis]]
* [[Fourier analysis]] and [[Fourier series]]
[[Category:Numerical analysis]]
[[Category:Fourier analysis]]
[[Category:Linear algebra]]
[[Category:Numerical linear algebra]]
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