Plancherel theorem ( $∥ f ∥_{2} = (2 π)^{n /2} ∥ f ∥_{2}$ )

For the unnormalized angular-frequency transform,

\int_{R^{n}} ∣ f (ξ) ∣^{2} d ξ = (2 π)^{n} \int_{R^{n}} ∣ f (x) ∣^{2} d x .

After normalization, the Fourier transform extends unitarily to

L^{2}

. For integrable

f

and

f

, inversion and Fubini give the identity directly.

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