Basis function

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).

Monomial basis for Cω
The monomial basis for the vector space of analytic functions is given by $$\{x^n \mid n\in\N\}.$$

This basis is used in Taylor series, amongst others.

Monomial basis for polynomials
The monomial basis also forms a basis for the vector space of polynomials. After all, every polynomial can be written as $$a_0 + a_1x^1 + a_2x^2 + \cdots + a_n x^n$$ for some $$n \in \mathbb{N}$$, which is a linear combination of monomials.

Fourier basis for L2[0,1]
Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions on a bounded domain. As a particular example, the collection $$\{\sqrt{2}\sin(2\pi n x) \mid n \in \N \} \cup \{\sqrt{2} \cos(2\pi n x) \mid n \in \N \} \cup \{1\}$$ forms a basis for L2[0,1].