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# Basis function

In mathematics, particularly numerical analysis, a basis function is an element of the 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"2} as a basis. Every quadratic can be written as "a"1+"bt"+"ct"2, that is, as a linear combination of the basis functions 1, "t", and "t"2. 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 basis.

Fourier basis

Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions. As a particular example, the collection::$\left\{sin\left(npi x\right) ; | ; ninmathbb\left\{Z\right\} ; ext\left\{and\right\} ; ngeq 1\right\} cup \left\{cos\left(npi x\right) ; | ; ninmathbb\left\{Z\right\} ; ext\left\{and\right\} ; ngeq 0\right\}$forms a basis for L2(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

* Orthogonal polynomials
* shape functions in the Galerkin method and finite element analysis
* Fourier analysis and Fourier series

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