Basis function

 ( Numerical Analysis ) 

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).

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Examples

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Monomial basis for C^ω The monomial basis for the vector space of analytic functions is given by $$\backslash \{x^n\; \backslash mid\; n\backslash in\backslash N\backslash \}.$$

This basis is used in Taylor series, amongst others.

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Monomial basis for polynomials The monomial basis also forms a basis for the vector space of . After all, every polynomial can be written as $a\_0\; +\; a\_1x^1\; +\; a\_2x^2\; +\; \backslash cdots\; +\; a\_n\; x^n$ for some $n\; \backslash in\; \backslash mathbb\{N\}$, which is a linear combination of monomials.

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Fourier basis for L²0,1 Sines and cosines form an (orthonormality) Schauder basis for square-integrable functions on a bounded domain. As a particular example, the collection $$\backslash \{\backslash sqrt\{2\}\backslash sin(2\backslash pi\; n\; x)\; \backslash mid\; n\; \backslash in\; \backslash N\; \backslash \}\; \backslash cup\; \backslash \{\backslash sqrt\{2\}\; \backslash cos(2\backslash pi\; n\; x)\; \backslash mid\; n\; \backslash in\; \backslash N\; \backslash \}\; \backslash cup\; \backslash \{1\backslash \}$$ forms a basis for

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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

• Kiyosi Itô (1993). 9780262590204, MIT Press. ISBN 9780262590204

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