Basis function
Template:Short description Template:Multiple issues 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 Cω
[edit]The monomial basis for the vector space of analytic functions is given by <math display="block">\{x^n \mid n\in\N\}.</math>
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 <math>a_0 + a_1x^1 + a_2x^2 + \cdots + a_n x^n</math> for some <math>n \in \mathbb{N}</math>, which is a linear combination of monomials.
Fourier basis for L2[0,1]
[edit]Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions on a bounded domain. As a particular example, the collection <math display="block">\{\sqrt{2}\sin(2\pi n x) \mid n \in \N \} \cup \{\sqrt{2} \cos(2\pi n x) \mid n \in \N \} \cup \{1\}</math> forms a basis for L2[0,1].
See also
[edit]- 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
References
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