Plancherel theorem

Remarks

Proof idea: for Schwartz functions the identity follows from the multiplication formula $\int \hat f \, \overline{\hat g} = \int f \, \overline{g}$ and Fourier inversion. Schwartz functions are dense in $L^1 \cap L^2$ for the norm $\norm{f}_1 + \norm{f}_2$, which gives part 1. Since $L^1 \cap L^2$ is dense in $L^2$, the isometry extends uniquely to $L^2$. The extension is onto because its range is closed and contains the Schwartz space.

For $f \in L^2$ the transform is computed as the $L^2$ limit $\mathcal F f = \lim_{R \to \infty} \int_{\abs{x} \le R} f(x) e^{-2\pi i x \cdot \xi} \, dx$, because the integral itself may diverge. The inner product is also preserved: $\int \mathcal F f \, \overline{\mathcal F g} = \int f \, \overline{g}$ for $f, g \in L^2(\R^n)$.

Example: for $f = \mathbf{1}_{[-1/2, 1/2]}$ on $\R$, $\hat f(\xi) = \frac{\sin \pi \xi}{\pi \xi}$ and $\int \abs{f}^2 = 1$. So the theorem gives $\int_\R \left( \frac{\sin \pi \xi}{\pi \xi} \right)^2 d\xi = 1$.

The analogue for Fourier series is Parseval's identity. The theorem generalizes to locally compact abelian groups and, with representations in place of characters, to more general groups.

Going deeper

For a student

Plancherel's theorem makes $L^2$ the natural home of the Fourier transform. It is the starting point of the $L^p$ theory, through interpolation with the trivial bound from $L^1$ to $L^\infty$. In quantum mechanics it says that the position and momentum representations of a wave function carry the same total probability. It also underlies the definition of Sobolev spaces $H^s$ by weights on the Fourier side.