basis function
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.
Examples
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.
Fourier basis
Sines and cosines form an (orthonormal) basis for square-integrable functions. As a particular example, the collection:
- Failed to parse (unknown function\text): \{\sin(n\pi x) \; | \; n\in\mathbb{Z} \; \text{and} \; n\geq 1\} \cup \{\cos(n\pi x) \; | \; n\in\mathbb{Z} \; \text{and} \; n\geq 0\}
forms a basis for L2(0,1).
References
- Ito, Kiyosi (1993). Encyclopedic Dictionary of Mathematics, 2nd ed., MIT Press, p. 1141. ISBN 0262590204.
See also
- Orthogonal polynomials
- Radial basis function
- shape functions in the Galerkin method and finite element analysis
- Fourier analysis and Fourier series
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