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Orthogonal Functions


Two functions f(x) and g(x) are orthogonal over the interval a<=x<=b with weighting function w(x) if

 <f(x)|g(x)>=int_a^bf(x)g(x)w(x)dx=0.
(1)

If, in addition,

int_a^b[f(x)]^2w(x)dx=1
(2)
int_a^b[g(x)]^2w(x)dx=1,
(3)

the functions f(x) and g(x) are said to be orthonormal.


See also

Complete Biorthogonal System, Complete Orthogonal System, Complete Set of Functions, Orthogonal Polynomials, Orthonormal Functions

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Cite this as:

Weisstein, Eric W. "Orthogonal Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrthogonalFunctions.html

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