# Basis function

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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).

## ExamplesEdit

### Monomial basis for Edit

The monomial basis is given by

This basis is used in amongst others Taylor series.

### Monomial basis for PolynomialsEdit

The monomial basis also forms a basis for the polynomials. After all, every polynomial can be written as , which is a linear combination of monomials.

### Fourier basis for Edit

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

forms a basis for L^{2}[0,1].

## ReferencesEdit

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