Parseval

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

In the same article, Kwapień proved that the validity of a Banach-valued Parseval's theorem for the Fourier transform characterizes Banach spaces isomorphic to Hilbert spaces.

Parseval's theorem was proved only for Fourier series, and was first proved by Liapounoff.

Plancherel theorem is a special case of the Parseval's theorem and states: : These theorems are also equivalent to the unitary condition below.

Such properties include the completeness, orthogonality, Plancherel/Parseval, periodicity, shift, convolution, and unitarity properties above, as well as many FFT algorithms.

There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum.

This allows us to extend Pythagoras' theorem to infinitely many summands: *Parseval's identity: Suppose V is a complete inner product space.