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Complete Biorthogonal System

A set of functions {f_1(n,x),f_2(n,x)} is termed a complete biorthogonal system in the closed interval R if, they are biorthogonal, i.e.,

int_Rf_1(m,x)f_1(n,x)dx=c_mdelta_(mn)
(1)
int_Rf_2(m,x)f_2(n,x)dx=d_mdelta_(mn)
(2)
int_Rf_1(m,x)f_2(n,x)dx=0
(3)
int_Rf_1(m,x)dx=0
(4)
int_Rf_2(m,x)dx=0
(5)

and complete.

A complete biorthogonal system has a very special type of generalized Fourier series. The prototypical example of a complete biorthogonal system is {sin(nx),cos(nx)}_(n=0)^infty over R=[-pi,pi], which can be used as a basis for constructing "the" Fourier series of an arbitrary function.

See also

Complete Orthogonal System, Fourier Series, Generalized Fourier Series

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Cite this as:

Weisstein, Eric W. "Complete Biorthogonal System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CompleteBiorthogonalSystem.html

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