Plancherel theorem

The Plancherel theorem is the Fourier transform's version of energy conservation, and the continuous counterpart of Parseval's identity for series. It says: the total energy of a function equals the total energy of its transform. Decomposing a signal into frequencies neither creates nor destroys energy.

Precisely, with the convention f-hat(xi) = integral of f(x) e^{-2*pi*i*x*xi} dx, for every f in L2 of the real line the transform f-hat is also in L2 and integral of |f(x)|^2 dx = integral of |f-hat(xi)|^2 dxi. There is a genuine technical point hidden here: the defining integral for f-hat converges absolutely only when f is integrable, but a general L2 function need not be integrable. So one first defines the transform on the nice functions (e.g. integrable and square-integrable) and then extends it to all of L2 by continuity, using exactly the norm-preservation as the tool that makes the extension well-defined.

The upshot is that the Fourier transform is a unitary operator on the Hilbert space L2 — it is a rotation of infinite-dimensional space that preserves all lengths and angles. Its inverse is the inverse transform, and inner products are preserved too (the polarized form: integral of f times conjugate g = integral of f-hat times conjugate g-hat). This unitarity is the structural backbone on which much of harmonic analysis and quantum mechanics rests.