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# Basis function

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 Title: Basis function Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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 {(t−1)(t−2)/2, −t(t−2), t(t−1)/2} is another basis for quadratic polynomials, called the Lagrange basis. The first three Chebyshev polynomials form yet another basis

### Fourier basis

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

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

forms a basis for L2(0,1).