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Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem.
More generally, given an abelian locally compact groupG with Pontryagin dualG^, Parseval's theorem says the Pontryagin-Fourier transform is a unitary operator between Hilbert spaces L2(G) and L2(G^) (with integration being against the appropriately scaled Haar measures on the two groups.) When G is the unit circleT, G^ is the integers and this is the case discussed above. When G is the real line , G^ is also and the unitary transform is the Fourier transform on the real line. When G is the cyclic groupZn, again it is self-dual and the Pontryagin-Fourier transform is what is called discrete Fourier transform in applied contexts.
Parseval's theorem can also be expressed as follows:
Suppose is a square-integrable function over (i.e., and are integrable on that interval), with the Fourier series
^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).
^Rayleigh, J.W.S. (1889) "On the character of the complete radiation at a given temperature," Philosophical Magazine, vol. 27, pages 460-469. Available on-line here.
^Plancherel, Michel (1910) "Contribution à l'etude de la representation d'une fonction arbitraire par les integrales définies," Rendiconti del Circolo Matematico di Palermo, vol. 30, pages 298-335.
^Arthur E. Danese (1965). Advanced Calculus. 1. Boston, MA: Allyn and Bacon, Inc. p. 439.