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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 cn include:

Bessel's Inequality

Parseval's Theorem

If Φ is a complete set,

See also:
orthonormal basis, orthogonal, square-integrable.