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.

Table of contents
- L2 density from a square-integrable Fourier transform Plancherel theorem

L2 density from a square-integrable Fourier transform ( $μ \in L^{2} ⟹ d μ = g d x, g \in L^{2}$ )

If a finite measure on

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

g d x

. This establishes absolute continuity, rather than presuming it. For a positive measure,

g \geq 0

and

g \in L^{1}

as well. Multiplication by

f \in L^{\infty} (μ)

yields

∥ f d μ ∥_{2} \leq ∥ μ ∥_{2} ∥ f ∥_{L^{\infty} (μ)}

.

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Plancherel theorem by

Wikipedia Bot  1

The Plancherel theorem is a fundamental result in the field of harmonic analysis, particularly in the context of Fourier transforms and Fourier series. It establishes an important relationship between the \( L^2 \) spaces of functions and distributions, indicating that the Fourier transform is an isometry on these spaces.

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Plancherel theorem by

Ciro Santilli  40 Updated 2025-07-16

Some sources say that this is just the part that says that the norm of a

L^{2}

function is the same as the norm of its Fourier transform.

Others say that this theorem actually says that the Fourier transform is bijective.

The comment at math.stackexchange.com/questions/446870/bijectiveness-injectiveness-and-surjectiveness-of-fourier-transformation-define/1235725#1235725 may be of interest, it says that the bijection statement is an easy consequence from the norm one, thus the confusion.

TODO does it require it to be in

l^{1}

as well? Wikipedia en.wikipedia.org/w/index.php?title=Plancherel_theorem&oldid=987110841 says yes, but courses.maths.ox.ac.uk/node/view_material/53981 does not mention it.

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