Basis function

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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<sup>ω</sup>''=

The monomial basis for the vector space of analytic functions is given by

$\{x^n \mid n\in\N\}.$

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 $a_0 + a_1x^1 + a_2x^2 + \cdots + a_n x^n$ for some $n \in \mathbb{N}$ , which is a linear combination of monomials.

=Fourier basis for ''L''<sup>2</sup>[0,1]=

Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions on a bounded domain. As a particular example, the collection

$\{\sqrt{2}\sin(2\pi n x) \mid n \in \N \} \cup \{\sqrt{2} \cos(2\pi n x) \mid n \in \N \} \cup \{1\}$

forms a basis for L²[0,1].

References

{{cite book |last=Itô |first=Kiyosi |title=Encyclopedic Dictionary of Mathematics |edition=2nd |year=1993 |publisher=MIT Press |isbn=0-262-59020-4 | page=1141}}

Category:Numerical analysis

Category:Fourier analysis

Category:Linear algebra

Category:Numerical linear algebra

Category:Types of functions