Shannon sampling theorem

The following is the version of Nyquist-Shannon sampling theorem due to Kluvanek.

Let \(G\) be a locally compact Abelian group. Let \(H \subseteq G\) be a discrete subgroup and let \(\Omega \subseteq \widehat{G}\) be a compact set which is often called as the band limit. We assume that \(\Omega\) is a fundamental domain of \(\widehat{G}/H^{\perp}\) where we define \[\begin{equation} H^{\perp} = \{\gamma \in \widehat{G} \mid \gamma(h)=1, \forall h \in H\} \subseteq \widehat{G}. \end{equation}\]

Pontryagin duality tells us that \[\begin{equation} \widehat{ \widehat{G}/H^{\perp}} \simeq H \end{equation}\]

We define a Hilbert space \[\begin{equation} \mathcal{H} = L^{2}(\Omega, \mathrm{vol}(\Omega)^{-1} \mathrm{d}\gamma ), \end{equation}\] with the inner product \[\begin{equation} \langle \varphi, \psi\rangle_{\mathcal{H}} = \mathrm{vol}(\Omega) ^{-1}\int_{\Omega}^{} \varphi(\gamma) \overline{\psi(\gamma) } \mathrm{d}\gamma \end{equation}\]

One defines \[\begin{equation} \pi: G \rightarrow \mathcal{U}(\mathcal{H}) \end{equation}\] via the twisting action \[\begin{equation} [\pi(x) \varphi] (\gamma) = \gamma(x) \varphi(\gamma). \end{equation}\]

We define \[\begin{equation} \Phi(x) = \langle \pi(x) 1 , 1\rangle = \mathrm{vol}(\Omega)^{-1} \int_{\Omega}^{} \gamma(x) \mathrm{d}\gamma \end{equation}\] Here \(1 \in \mathcal{H}\) is the constant \(1\) function, which exists by compactness of \(\Omega\).

The function \(\Phi: G \rightarrow \mathbb{C}\) satisfies the following \[\begin{equation} \Phi(x) = \begin{cases} 1 & \text{ if } x=0\\ 0 & \text{ if }x \in H \setminus \{0\} \end{cases} \end{equation}\]

Assume \(x \in H\). Also assume without loss of generality that \(\mathrm{d}\gamma\) is a probability measure. Then we have \[\begin{align} \Phi(x) = & \int_{\Omega}^{} [\pi(x) 1](\gamma) \cdot 1(\gamma.html) \mathrm{d}\gamma \\ = & \int_{\Omega}^{} \gamma(x) \mathrm{d}\gamma \\ = & \int_{\widehat{G}/H^{\perp}}^{} \gamma(x) \mathrm{d}\gamma. \end{align}\] Here we are able to change the integral from \(\Omega\) to \(\widehat{G}/H^{\perp}\) because \[\begin{equation} \gamma(x) = (\gamma \eta)(x) \forall \eta \in H^{\perp} (\text{ since } x \in H \implies \eta(x)=1). \end{equation}\] Then we conclude from the fact that by Pontryagin duality, \(x\) is a character on \(\widehat{G}/H^{\perp}\).

From orthogonality of characters, we also get that \[\begin{equation} \mathcal{B} = \{ \pi(-h) \cdot 1\}_{h \in H} \text{ is an orthonormal basis of } \mathcal{H}. \end{equation}\]

Define the Fourier transform via \[\begin{equation} \widehat{f}(\xi) = \int_{G}^{} f(x) \xi(-x) \mathrm{d}\mu (x), \end{equation}\] where \(\mathrm{d}\mu\) is some Haar measure on \(G\).

Suppose \(\widehat{f} \in \mathcal{H}\). Then we have \[\begin{align} \langle \widehat{f}, \pi(-x ) 1 \rangle_{\mathcal{H}} & = \mathrm{vol}(\Omega)^{-1} \int_{\Omega}^{} \widehat{f}(\gamma) \overline{\gamma}(-x) \mathrm{d}\gamma \\ = \mathrm{vol}(\Omega)^{-1} \int_{\Omega}^{} \widehat{f}(\gamma) {\gamma}(x) \mathrm{d}\gamma \\ = \mathrm{vol}(\Omega)^{-1} f(x) \end{align}\] Here we used the Fourier inversion formula. The compatibility of \(\mathrm{d}\mu\) and \(\mathrm{d}\gamma\) is essential for the Fourier inversion formula.

By using the orthonormal basis, we get \[\begin{align} \widehat{f} = & \sum_{h \in H}^{} \langle \widehat{f}, \pi(-h) 1 \rangle _{\mathcal{H}} \pi(-h) 1 \\ = & \sum_{h \in H}^{} \mathrm{vol}(\Omega)^{-1} f(h) \pi(-h) 1 \end{align}\]

We finish by noting that \[\begin{align} \mathrm{vol}(\Omega)^{-1}f(x) & = \langle \widehat{f} , \pi (-x) 1 \rangle_{ \mathcal{H}} \\ & = \langle \sum_{h \in H}^{} \frac{f(h)}{\mathrm{vol}(\Omega)} \pi(-h) 1 , \pi(-x) 1 \rangle \\ & = \sum_{h \in H}^{} \frac{f(h)}{\mathrm{vol}(\Omega)} \Phi( x- h). \end{align}\]

Given a locally compact Abelian group \(G\) and a discrete cocompact subgroup \(H\) and a fundamental domain \(\Omega \subseteq \widehat{G}\) of \(\widehat{G}/H^{\perp}\), one can recover an \(L^{2}\)-function \(f\) whose Fourier transform is supported on \(\Omega\) from its values on \(H\).