Basis function

Short description: Element of a basis for a function space

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 [math]\displaystyle{ \{x^n \mid n\in\N\}. }[/math]

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

Fourier basis for L²[0,1]

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

• 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

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