In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is 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.[1] It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh.[2]
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.[3]
As is the case with the middle terms in this example, many terms will integrate to over a full period of length (see harmonics):
More generally, if and are instead two complex-valued functions on of period that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series
and
respectively. Then
(Eq.2)
Even 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 spacesL2(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
where represents the continuous Fourier transform (in non-unitary form) of , and is frequency in radians per second.
The interpretation of this form of the theorem is that the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.
^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. Vol. 1. Boston, MA: Allyn and Bacon, Inc. p. 439.
^Wilfred Kaplan (1991). Advanced Calculus (4th ed.). Reading, MA: Addison Wesley. p. 519. ISBN 0-201-57888-3.
^Georgi P. Tolstov (1962). Fourier Series. Translated by Silverman, Richard. Englewood Cliffs, NJ: Prentice-Hall, Inc. p. 119.