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Orthonormal Functions


A pair of functions phi_i(x) and phi_j(x) are orthonormal if they are orthogonal and each normalized so that

int_a^b[phi_i(x)]^2w(x)dx=1
(1)
int_a^b[phi_j(x)]^2w(x)dx=1.
(2)

These two conditions can be succinctly written as

 int_a^bphi_i(x)phi_j(x)w(x)dx=delta_(ij),
(3)

where w(x) is a weighting function and delta_(ij) is the Kronecker delta.


See also

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

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Cite this as:

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

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