# 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

### Monomial basis for ${\displaystyle C^{\omega }}$

The monomial basis is given by

${\displaystyle \{x^{n}\mid n\in \mathbb {N} \}.}$

This basis is used in amongst others Taylor series.

### Monomial basis for Polynomials

The monomial basis also forms a basis for the polynomials. After all, every polynomial can be written as ${\displaystyle a_{0}+a_{1}x^{1}+a_{2}x^{2}+\dots }$ , which is a linear combination of monomials.

### Fourier basis for ${\displaystyle L^{2}[0,1]}$

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

${\displaystyle \{{\sqrt {2}}\sin(2\pi nx)\;|\;n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\;|\;n\in \mathbb {N} \}\cup \{1\}}$

forms a basis for L2[0,1].

## References

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