Basis function

In mathematics, a basis function is an element of a particular basis for a function space. Every continuous 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

Polynomial bases

The collection of quadratic polynomials with real coefficients has {1, t, t²} as a basis. Every quadratic polynomial can be written as a1+bt+ct², that is, as a linear combination of the basis functions 1, t, and t². The set {(1/2)(t-1)(t-2), -t(t-2), (1/2)t(t-1)} is another basis for quadratic polynomials, called the Lagrange basis.

Fourier basis

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

$\{\sqrt{2}\sin(n\pi x) \; | \; n\in\mathbb{Z} \; \text{and} \; n\geq 1\} \cup \{\sqrt{2} \cos(n\pi x) \; | \; n\in\mathbb{Z} \; \text{and} \; n\geq 1\} \cup\{1\}$

forms a basis for L²(0,1).

References

• Ito, Kiyoshi (1993). Encyclopedic Dictionary of Mathematics (2nd ed. ed.). MIT Press. pp. 1141. ISBN 0262590204. 

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