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Concept
Parseval's theorem
Summary
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 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.
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.
Suppose that and are 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
where is the imaginary unit and horizontal bars indicate complex conjugation. Substituting and :
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, given an abelian locally compact group G with Pontryagin dual G^, 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 circle T, 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 group Zn, 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
Then
In electrical engineering, Parseval's theorem is often written as:
where represents the continuous Fourier transform (in normalized, unitary form) of , and is frequency in radians per second.
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