the Plancherel theorem
/ plahn-shuh-REL /
When you split a signal into its frequencies, do you lose anything? The Plancherel theorem answers no in the strongest possible sense: the total 'energy' of a signal, measured as the integral of its square, is exactly the same whether you compute it in physical space or in frequency space. The transform merely repackages the same energy across different frequencies; it neither creates nor destroys it.
The statement is a single clean identity: integral from -infinity to infinity of |f(x)|^2 dx equals (1/(2*pi)) integral from -infinity to infinity of |f-hat(xi)|^2 dxi (the 2*pi placement depends on your convention). The left side adds up the squared size of the signal across positions; the right side adds up the squared size of its spectrum across frequencies; the two totals agree. The closely related Parseval form says the same for the inner product of two functions, not just a function with itself: integral of f times the conjugate of g equals (1/(2*pi)) integral of f-hat times the conjugate of g-hat. In the language of linear algebra, the Fourier transform is a rotation (a unitary map) — it preserves lengths and angles.
Why it matters for PDEs: it is the natural way to measure the size of a solution. For the heat equation u-hat(xi, t) = f-hat(xi) e^(-k*xi^2 t), the factor e^(-k*xi^2 t) is between 0 and 1, so each |u-hat|^2 only shrinks; by Plancherel the integral of u^2 can only decrease in time. That is an energy estimate — a clean proof that diffusion damps everything — gotten for free from the transform. The same identity underlies stability and uniqueness arguments throughout the subject and is the bridge to the L^2 (Sobolev) spaces in which modern PDE theory is set.
Take f(x) = e^(-|x|). Its squared integral in space is integral of e^(-2|x|) dx = 1. Its transform is f-hat(xi) = 2/(1+xi^2), and (1/(2*pi)) integral of (2/(1+xi^2))^2 dxi also equals 1. Two very different-looking integrals give the same number — that is Plancherel doing its bookkeeping.
The transform is a rotation: it preserves total energy, position-side equals frequency-side.
Many books call the same result Parseval's theorem; the historical split is that Parseval first stated it for Fourier series and Plancherel for the Fourier integral, but in practice the names are used interchangeably.