Editing Generalized Fourier series (section)

== Coefficient theorems ==
Some theorems on the series' coefficients <math>c_n</math> include:

=== [[Bessel's inequality]] ===
'''Bessel's inequality''' is a statement about the coefficients of an element <math>x</math> in a [[Hilbert space]] with respect to an [[orthonormal]] [[sequence]]. The inequality was derived by [[Frederic Bessel|F.W. Bessel]] in 1828:<ref>{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Bessel_inequality|title = Bessel inequality - Encyclopedia of Mathematics}}</ref>
:<math>\sum_{n=0}^\infty |c_n|^2\leq\int_a^b|f(x)|^2w(x)\,dx.</math>

=== [[Parseval's theorem]] ===
'''Parseval's theorem''' usually refers to the result that the [[Fourier transform]] is [[Unitary operator|unitary]]; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.<ref>Parseval des Chênes, Marc-Antoine Mémoire sur les séries et sur l'intégration complète d'une équation aux différences partielles linéaire du second ordre, à coefficients constants" presented before the Académie des Sciences (Paris) on 5 April 1799. This article was published in ''Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savants, et lus dans ses assemblées. Sciences, mathématiques et physiques. (Savants étrangers.)'', vol. 1, pages 638–648 (1806).</ref>

If Φ is a complete basis, then:
:<math> \sum_{n=0}^\infty |c_n|^2 = \int_a^b |f(x)|^2w(x)\, dx.</math>