# Basis function

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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:

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

forms a basis for L2(0,1).

## References

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

## See also

Basis (linear algebra) (Hamel basis) Schauder basis (in a Banach space) Dual basis Biorthogonal system (Markushevich basis)
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