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Basis function

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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[edit]

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

This basis is used in Taylor series, amongst others.

Monomial basis for polynomials[edit]

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.

Fourier basis for L2[0,1][edit]

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 L2[0,1].

See also[edit]


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