# Generalized Fourier series

In

mathematical analysis, there are many potentially useful generalizations of

Fourier series. For a set of

square-integrable,

pairwise-orthogonal (with respect to some

weight function *w*(

*x*)) functions

the

**generalized Fourier series** of a

square-integrable function

*f*:[

*a*,

*b*] → C is

where the coefficients are determined by

The relation becomes equality if Φ is a complete set, i.e., an

orthonormal basis of the space of all square-integrable functions on [

*a*,

*b*], as opposed to a smaller orthonormal set, provided the convergence of the series is understood to be convergence in mean square and not necessarily pointwise convergence, nor convergence

almost everywhere.

Some theorems on the coefficients *c*_{n} include:

## Parseval's Theorem

If Φ is a complete set,

See also: orthonormal basis,

orthogonal,

square-integrable.