Generalized Fourier series
In
mathematical analysis, theremany potentially useful generalizations
Fourier series. Forset
square-integrable,
pairwise-orthogonal (with respectsome
weight function w(
x)) functions
the
generalized Fourier series of
square-integrable function
f:[
a,
b] → C is
wherecoefficientsdetermined by
The relation becomes equality if Φ iscomplete set, i.e., an
orthonormal basis ofspaceall square-integrable functions on [
a,
b], as opposed tosmaller orthonormal set, providedconvergence ofseriesunderstoodbe convergencemean squarenot necessarily pointwise convergence, nor convergence
almost everywhere.
Some theorems oncoefficients cn include:
Bessel's Inequality
Parseval's Theorem
If Φ iscomplete set,
See also:
orthonormal basis,
orthogonal,
square-integrable.
