Pontryagin duality

In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. It places in a unified context a number of observations about functions on the real line or on finite abelian groups:

Plancherel and Fourier inversion theorems

As we have stated, the dual group of a locally compact abelian group is a locally compact abelian group in its own right and thus has a Haar measure, or more precisely a whole family of scale-related Haar measures.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Theorem. There is a scaling of Haar measure on the dual group so that the Fourier transform restricted to continuous functions of compact support on G, is an isometric linear map. It has a unique extension to a unitary operator

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: mathcal{F}: L^2_mu(G) ightarrow L^2_ u(widehat{G})

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

where u is the Haar measure on the dual group.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Note that for non-compact locally compact groups G the space L1(G) does not contain L2(G), so one has to resort to some technical trick such as restricting to a dense subspace.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Following the Loomis reference below, we say that Haar measures on G and G^ are associated if and only if the Fourier inversion formula holds. The unitary character of the Fourier transform implies:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: int_G |f(x)|^2 d mu(x) = int_{widehat{G}} |widehat{f}(chi)|^2 d u(chi)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

for every continuous complex-valued function of compact support on G.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It is the unitary extension of the Fourier transform which we consider to be the Fourier transform on the space of square integrable functions. The dual group also has an inverse Fourier transform in its own right; it can be characterized as the inverse (or adjoint, since it is unitary) of the Fourier transform. This is the content of the Fourier inversion formula which follows.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Theorem. The adjoint of the Fourier transform restricted to continuous functions of compact support is the inverse Fourier transform

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: L^2_ u(widehat{G}) ightarrow L^2_mu(G)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

where the measures on G and G^ are associated.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In the case of G = Rn, we have G′ = Rn and we recover the ordinary Fourier transform on the Rn by taking

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: mu = (2 pi)^{-n/2} imes mbox{Lebesgue measure}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: u = (2 pi)^{-n/2} imes mbox{Lebesgue measure}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In the case G = T, the dual group G′ is naturally isomorphic to the group of integers Z and the above operator F specializes to the computation of coefficients of Fourier series of periodic functions.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If G is a finite group, we recover the discrete Fourier transform. Note that this case is very easy to prove directly.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~