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Basis function

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(Redirected from Basis functions) Element of a basis for a function space
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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

Monomial basis for C

The monomial basis for the vector space of analytic functions is given by { x n ∣ n ∈ N } . {\displaystyle \{x^{n}\mid n\in \mathbb {N} \}.}

This basis is used in Taylor series, amongst others.

Monomial basis for polynomials

The monomial basis also forms a basis for the vector space of polynomials. After all, every polynomial can be written as a 0 + a 1 x 1 + a 2 x 2 + ⋯ + a n x n {\displaystyle a_{0}+a_{1}x^{1}+a_{2}x^{2}+\cdots +a_{n}x^{n}} for some n ∈ N {\displaystyle n\in \mathbb {N} } , which is a linear combination of monomials.

Fourier basis for L

Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions on a bounded domain. As a particular example, the collection { 2 sin ⁡ ( 2 π n x ) ∣ n ∈ N } ∪ { 2 cos ⁡ ( 2 π n x ) ∣ n ∈ N } ∪ { 1 } {\displaystyle \{{\sqrt {2}}\sin(2\pi nx)\mid n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\mid n\in \mathbb {N} \}\cup \{1\}} forms a basis for L.

See also

References

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