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.,
[画像:int_Rf_1(m,x)f_1(n,x)dx] = c_mdelta_(mn)
(1)
[画像:int_Rf_2(m,x)f_2(n,x)dx] = d_mdelta_(mn)
(2)
[画像:int_Rf_1(m,x)f_2(n,x)dx] =
(3)
[画像:int_Rf_1(m,x)dx] =
(4)
[画像:int_Rf_2(m,x)dx] =
(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 SeriesExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Complete Biorthogonal System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CompleteBiorthogonalSystem.html