Basis function

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In mathematics, particularly numerical analysis, a basis function is an element of the basis for a function space. The term is a degeneration of the term basis vector for a more general vector space; that is, each function in the function space can be represented as a linear combination of the basis functions.

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[edit] Examples

[edit] Polynomial bases

The collection of quadratic polynomials with real coefficients has {1, t, t2} as a basis. Every quadratic 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.

[edit] Fourier basis

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

\{\sin(n\pi x) \; | \; n\in\mathbb{Z} \; \text{and} \; n\geq 1\} \cup \{\cos(n\pi x) \; | \; n\in\mathbb{Z} \; \text{and} \; n\geq 0\}

forms a basis for L2(0,1).

[edit] References

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

[edit] See also

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