This post’ll be pretty quick—the Plancherel theorem, a basic result on Fourier transforms, is a quick corollary of what I’ve already talked about.

We have shown that the Fourier transform is an isomorphism of {\mathcal{S}} onto itself; the inverse is given by the inverse Fourier transform. The next step is to extend this to an isometry of {L^2} onto itself. Since {\mathcal{S}} is dense in {L^2}, it will be sufficient to show that

\displaystyle ||f||_2 = ||\hat{f}||_2

for {f \in \mathcal{S}}. We will do this by proving the identity

\displaystyle (\hat{f},g) = (f, \tilde{g})

for {f,g \in L^2}. This is a simple computation:

\displaystyle (\hat{f},g) = \iint f(y) e^{-2 \pi i x.y} \overline{ g(x)} dy dx = (f, \tilde{g}).

In other words, the Fourier transform and its inverse are adjoints. If we take {g = \hat{f}} and use the inversion formula, it becomes clear that the Fourier transform preserves the {L^2}-norm, whence follows

Theorem 1 (Plancherel) The Fourier transform extends to an isometry of {L^2} onto itself.


Incidentally, for a {L^2} function {f}, it is not necessarily true that

\displaystyle \hat{f}(x) = \int f(y) e^{-2 \pi i x.y} dy

because that integral need not exist. It is, however, true that the integral will exist almost everywhere in a “principal value” sense, which we do not need to bother with here.

In a sense, this is a continuous analog of the Parseval theorem, which states that the Fourier coefficient map from {L^2 \rightarrow l^2} (for {l^2} the space of double-sided, square-summable sequences) is an isometry.