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### basis function

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ARTICLE ORIGINS basis function
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{{more footnotes|date=March 2013}}In mathematics, a basis function is an element of a particular basis for a function space. Every continuous 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

### Polynomial bases

The base of a polynomial is the factored polynomial equation into a linear function.JOURNAL, 2007-08-01, Solutions of differential equations in a Bernstein polynomial basis,weblink Journal of Computational and Applied Mathematics, en, 205, 1, 272â€“280, 10.1016/j.cam.2006.05.002, 0377-0427,

### Fourier basis

Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions. As a particular example, the collection:
{sqrt{2}sin(2pi n x) ; | ; ninmathbb{N} } cup {sqrt{2} cos(2pi n x) ; | ; ninmathbb{N} } cup{1}
forms a basis for L2(0,1).

## References

• BOOK, Ito, Kiyoshi, Encyclopedic Dictionary of Mathematics, 2nd, 1993, MIT Press, 0-262-59020-4, 1141,

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