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).

ExamplesEdit

Monomial basis for  Edit

The monomial basis is given by

 

This basis is used in amongst others Taylor series.

Monomial basis for PolynomialsEdit

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

Fourier basis for  Edit

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

 

forms a basis for L2[0,1].

ReferencesEdit

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

See alsoEdit

ReferencesEdit