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. In finite-dimensional vector spaces, this representation is purely algebraic and involves only finitely many basis functions, whereas in infinite-dimensional settings it may take the form of an infinite series or another limiting process whose convergence depends on the topology of the space.

The choice of basis functions is not unique. Different bases can represent the same function space, but can have different properties that are useful for particular applications. For example, monomials are convenient for elementary polynomial calculations, trigonometric functions are useful for Fourier analysis, and locally supported functions are useful in numerical methods.

In numerical analysis and approximation theory, basis functions are also called blending functions, particularly in applications such as interpolation. In this application, a mixture of the basis functions provides an interpolating function, with the coefficients determined by the data being interpolated. Basis functions are also used extensively in finite element methods, splines, wavelets, and other approximation methods.