Fourier Basis

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^ω''

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

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 $n \in \mathbb$, which is a linear combination of monomials.

Fourier basis for ''L''² ,1/h2>

Sines and cosines form an ( orthonormal) Schauder basis for square-integrable functions on a bounded domain. As a particular example, the collection
$\ \cup \ \cup \$
forms a basis for ''L''² ,1

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

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

Numerical analysis
Fourier analysis
Linear algebra
Numerical linear algebra
Types of functions