Parseval's theorem shows that T is bounded from to with norm 1.

In physics and engineering, Parseval's theorem is often written as:

This expansion has the property, analogous to Parseval's theorem, that:

This is consistent with Parseval's theorem.

Parseval's theorem is a special case of the Plancherel theorem and states:

This follows from Parseval's theorem.

Another application of this Fourier series is to solve the Basel problem by using Parseval's theorem.

Parseval's theorem.

Hence Parseval's theorem easily shows that the Hilbert transform is bounded from to .

The RMS can be computed in the frequency domain, using Parseval's theorem.

For the special case , this implies that the length of a vector is preserved as well-this is just Parseval's theorem:

In this case, Parseval's theorem gives us an alternate expression for the energy of the signal in terms of its Fourier transform, :

All of the basic properties listed above hold for the 'n'-dimensional Fourier transform, as do Plancherel's and Parseval's theorem.

As a consequence of Parseval's theorem, one can prove that the signal energy is always equal to the summation across all frequency components of the signal's spectral energy density.

Parseval's theorem allows to solve this problem completely and obtain that a function m is an L(G) multiplier if and only if it is bounded and measurable.

It can be shown using Parseval's theorem and an isometry that approximating the discrete time rational transform is equivalent to approximating the 2-EPT density itself in the L-2 Norm sense.

The appropriate choice of scaling to achieve unitarity is , so that the energy in the physical domain will be the same as the energy in the Fourier domain, i.e., to satisfy Parseval's theorem.

Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics and engineering, the most general form of this property is more properly called the Plancherel theorem.

The unitarity of the Fourier transform is often called Parseval's theorem in science and engineering fields, based on an earlier (but less general) result that was used to prove the unitarity of the Fourier series.

One of the results of Fourier analysis is Parseval's theorem which states that the area under the energy spectral density curve is equal to the area under the square of the magnitude of the signal, the total energy:

The transforms are linear operators and, with proper normalization, are unitary as well (a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality) .

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.