Basis Functions

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 collection of quadratic polynomials with real coefficients has {1, t, t2} as a basis. Every quadratic polynomial can be written as a1+bt+ct2, that is, as a linear combination of the basis functions 1, t, and t2. The set {(1/2)(t-1)(t-2), -t(t-2), (1/2)t(t-1)} is another basis for quadratic polynomials, called the Lagrange basis.

Fourier basis

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

$\\left\{\sqrt\left\{2\right\}\sin\left(n\pi x\right) \; | \; n\in\mathbb\left\{N\right\} \\right\} \cup \\left\{\sqrt\left\{2\right\} \cos\left(n\pi x\right) \; | \; n\in\mathbb\left\{N\right\} \\right\} \cup\\left\{1\\right\}$

forms a basis for L2(0,1)Template:Dn.