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[edit]
Monomial basis for C^{ω}[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 L^{2}[0,1][edit]
Sines and cosines form an (orthonormal) Schauder basis for squareintegrable functions on a bounded domain. As a particular example, the collection
See also[edit]
 Basis (linear algebra) (Hamel basis)
 Schauder basis (in a Banach space)
 Dual basis
 Biorthogonal system (Markushevich basis)
 Orthonormal basis in an innerproduct space
 Orthogonal polynomials
 Fourier analysis and Fourier series
 Harmonic analysis
 Orthogonal wavelet
 Biorthogonal wavelet
 Radial basis function
 Finiteelements (bases)
 Functional analysis
 Approximation theory
 Numerical analysis
References[edit]
 Itô, Kiyosi (1993). Encyclopedic Dictionary of Mathematics (2nd ed.). MIT Press. p. 1141. ISBN 0262590204.