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.
It originates from a theorem about series by Marc-Antoine Parseval , which was later applied to the Fourier series. 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. When G is the cyclic group Z n , again it is self-dual and the Pontryagin—Fourier transform is what is called discrete Fourier transform in applied contexts.
Then   . In physics and engineering, Parseval's theorem is often written as:.
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. For discrete time signals , the theorem becomes:.
Alternatively, for the discrete Fourier transform DFT , the relation becomes:. Parseval's theorem is closely related to other mathematical results involving unitary transformations:.
From Wikipedia, the free encyclopedia. Available on-line here. Danese Advanced Calculus.
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Advanced Calculus 4th ed. Reading, MA: Addison Wesley.
Tolstov Fourier Series. Translated by Silverman, Richard. Categories : Theorems in Fourier analysis.