This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)
This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (March 2013) (Learn how and when to remove this message)
This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (September 2019) (Learn how and when to remove this message)
This article may need to be rewritten to comply with Wikipedia's quality standards. You can help. The talk page may contain suggestions. (September 2019)
(Learn how and when to remove this message)

In mathematics, a **basis function** is an element of a particular basis for a function space. Every 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).

The monomial basis for the vector space of analytic functions is given by

This basis is used in Taylor series, amongst others.

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

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

forms a basis for