Basis function
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In mathematics, particularly numerical analysis, a basis function is an element of the basis for a function space. The use of the term is analogous to basis vector for a vector space; that is, each function in the function space can be represented as a linear combination of the basis functions.
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[edit] Examples
[edit] Polynomial bases
The collection of quadratic polynomials with real coefficients has {1, t, t2} as a basis. Every quadratic can be written as a1+bt+ct2, that is, as a linear combination of the basis functions 1, t, and t2. The set {(1/2)(t-1)(t-2), -t(t-2), (1/2)t(t-1)} is another basis for quadratic polynomials, called the Lagrange basis.
[edit] Fourier basis
Sines and cosines form an (orthonormal) basis for square-integrable functions. As a particular example, the collection:
forms a basis for L2(0,1).
[edit] See also
- Orthogonal polynomials
- Radial basis function
- shape functions in the Galerkin method and finite element analysis
- Fourier analysis and Fourier series
