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).

Examples

Monomial basis for C^ω

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

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 $a_{0}+a_{1}x^{1}+a_{2}x^{2}+\cdots +a_{n}x^{n}$ 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 $\{{\sqrt {2}}\sin(2\pi nx)\mid n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\mid n\in \mathbb {N} \}\cup \{1\}$ forms a basis for L²[0,1].

References

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

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Basis function