# Basis function

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.[1]

### Fourier basis

Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions. As a particular example, the collection:

forms a basis for L^{2}(0,1).

## References

- Itô, Kiyosi (1993).
*Encyclopedic Dictionary of Mathematics*(2nd ed.). MIT Press. p. 1141. ISBN 0-262-59020-4.

## See also

## References

- Idrees Bhatti, M.; Bracken, P. (2007-08-01). "Solutions of differential equations in a Bernstein polynomial basis".
*Journal of Computational and Applied Mathematics*.**205**(1): 272–280. doi:10.1016/j.cam.2006.05.002. ISSN 0377-0427.