Plancherel theorem

In mathematics, the Plancherel theorem is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum.

A more precise formulation is that if a function is in both L(R) and L(R), then its Fourier transform is in L(R), and the Fourier transform map is an isometry with respect to the L norm. This implies that the Fourier transform map restricted to L(R) ∩ L(R) has a unique extension to a linear isometric map L(R) → L(R). This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of quadratically integrable functions.