Orthonormal Functions
A pair of functions phi_i(x) and phi_j(x) are orthonormal if they are orthogonal and each normalized so that
[画像:int_a^b[phi_i(x)]^2w(x)dx] = 1
![画像:int_a^b[phi_i(x)]^2w(x)dx](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/OrthonormalFunctions/Inline3.svg){kind=link}
(1)
[画像:int_a^b[phi_j(x)]^2w(x)dx] = 1.
![画像:int_a^b[phi_j(x)]^2w(x)dx](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/OrthonormalFunctions/Inline6.svg){kind=link}
(2)
These two conditions can be succinctly written as
where w(x) is a weighting function and delta_(ij) is the Kronecker delta.
See also
Complete Biorthogonal System, Complete Orthogonal System, Orthogonal Polynomials, Orthonormal Basis, Orthonormal VectorsExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Orthonormal Functions." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OrthonormalFunctions.html