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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
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 for some , which is a linear combination of monomials.
Fourier basis for L2[0,1]
forms a basis for L2[0,1].
- Itô, Kiyosi (1993). Encyclopedic Dictionary of Mathematics (2nd ed.). MIT Press. p. 1141. ISBN 0-262-59020-4.
- Basis (linear algebra) (Hamel basis)
- Schauder basis (in a Banach space)
- Dual basis
- Biorthogonal system (Markushevich basis)
- Orthonormal basis in an inner-product space
- Orthogonal polynomials
- Fourier analysis and Fourier series
- Harmonic analysis
- Orthogonal wavelet
- Biorthogonal wavelet
- Radial basis function
- Finite-elements (bases)
- Functional analysis
- Approximation theory
- Numerical analysis