Parseval-Plancherel theorem
/ par-suh-VAL, plahn-shuh-REL /
If you measure the total energy of a signal — say by adding up the square of its amplitude over all time — you should get the same answer whether you look at the signal itself or at its spectrum. Energy does not appear or vanish just because you changed your viewpoint from time to frequency. The Parseval-Plancherel theorem is the precise guarantee of this: the Fourier transform conserves energy, distributing the same total over the frequencies that it had over time.
In symbols, integral over x of |f(x)|^2 dx = (1/(2 pi)) integral over k of |F(k)|^2 dk (the constant depends on the convention; with the symmetric 1/sqrt(2 pi) normalization there is no extra factor and the two integrals are simply equal). More generally, for two functions, integral of f(x) conjugate-of-g(x) dx equals the same inner product of their transforms — the Fourier transform preserves inner products, so it is a unitary map. The quantity |F(k)|^2 is therefore read as the energy spectral density: integrating it over a band tells you how much of the signal's energy lives in that band.
This is the foundation of every energy and power calculation in the frequency domain. It says the spectrum is not just a bookkeeping device but a faithful redistribution of the same physical energy, which is why the power spectrum is a meaningful physical object in acoustics, optics, and communications. Mathematically, unitarity is what makes the Fourier transform the natural tool on the space of square-integrable functions, and it underlies the rigorous extension of the transform to functions (like a pure sinusoid) whose defining integral does not converge in the ordinary sense.
Sometimes a hard integral over x is easy over k. For the pulse with F(k) = 2 sin(k)/k, the unit pulse has energy integral over x of |f|^2 = 2, so Parseval gives (1/(2 pi)) integral over k of (2 sin(k)/k)^2 dk = 2, i.e. integral of (sin k / k)^2 dk = pi — evaluating a tricky integral by energy balance.
Energy conservation doubles as an integration trick: equate the energy in both domains to evaluate an integral one of them makes hard.
The constant relating the two integrals depends entirely on the chosen Fourier convention; quote Parseval's identity with the wrong normalization and your energy will be off by a factor of 2 pi.