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.

Fourier basis

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

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

forms a basis for L²(0,1).

References

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

See also

• 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