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).

Examples

Polynomial bases

The base of a polynomial is the factored polynomial equation into a linear function.[1]

Fourier basis

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

${\displaystyle \{{\sqrt {2}}\sin(2\pi nx)\;|\;n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\;|\;n\in \mathbb {N} \}\cup \{1\}}$

forms a basis for L2(0,1).

References

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