Plancherel theorem ($\Vert \hat{f}{\Vert }_2=(2\pi {)}^{n/2}\Vert f{\Vert }_2$)

For the unnormalized angular-frequency transform,After normalization, the Fourier transform extends unitarily to $L^2$. For integrable $f$ and $\hat{f}$, inversion and Fubini give the identity directly.

With the Fourier convention $\hat{f}(t)=\int f(x)e^{-itx}dx$, translation by $k$ multiplies the transform by $e^{ikt}$. The Plancherel theorem gives the displayed correlation identity for every square-integrable function, by extension from integrable square-integrable functions. Thus distinct integer translates are orthogonal exactly when the nonzero integer Fourier integrals of $|\hat{f}{|}^2$ vanish. The products in both integrals are integrable by the Cauchy-Schwarz inequality.

For $h\in C_c(G)$, apply Fourier inversion on a locally compact abelian group at the identity to $h\ast h^{\ast }$, whose transform is $|\hat{h}{|}^2$. This proves the displayed norm identity on $C_c(G)$. Density and completeness give a unique linear isometric extension to all $L^2(G)$. On $L^1\cap L^2$ it agrees almost everywhere with the integral Fourier transform on a locally compact abelian group.

If a finite measure on ${\mathbb{R}}^n$ has square-integrable Fourier transform, the Plancherel theorem gives an $L^2$ inverse transform $g$. Pairing with every Schwartz function and using Fourier inversion identifies the measure with $gdx$. This establishes absolute continuity, rather than presuming it. For a positive measure, $g\ge 0$ and $g\in L^1$ as well. Multiplication by $f\in L^{\infty }(\mu )$ yields $\Vert \widehat{fd\mu }{\Vert }_2\le \Vert \hat{\mu }{\Vert }_2\Vert f{\Vert }_{L^{\infty }(\mu )}$.