- Basis function
In
mathematics , particularlynumerical analysis , a basis function is an element of the basis for afunction space . The term is a degeneration of the term "basis vector" for a more generalvector space ; that is, each function in the function space can be represented as alinear combination of the basis functions.Examples
Polynomial bases
The collection of quadratic polynomials with real coefficients has {1, "t", "t"2} as a basis. Every quadratic can be written as "a"1+"bt"+"ct"2, that is, as a
linear combination of the basis functions 1, "t", and "t"2. 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::sin(npi x) ; | ; ninmathbb{Z} ; ext{and} ; ngeq 1} cup {cos(npi x) ; | ; ninmathbb{Z} ; ext{and} ; ngeq 0}forms a basis for L2(0,1).References
*cite book |last=Ito |first=Kiyosi |authorlink= |coauthors= |others= |title=Encyclopedic Dictionary of Mathematics |edition=2nd ed. |year=1993 |publisher=MIT Press |location= |id=ISBN 0262590204 | pages=p. 1141
See also
*
Orthogonal polynomials
*Radial basis function
* shape functions in theGalerkin method andfinite element analysis
*Fourier analysis andFourier series
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