## 2 Materials and Methods
Fourier analysis provides a bijective isometry between \(\ell^{2}\) and \(\mathcal{B}_{\sigma}^{2}\): defining \(\mathbf{x}_{\mathbf{f}}[k]:=\mathbf{f}(k\pi/\sigma)\), \(k\in\mathbb{Z}\), we have  
\[\big{[}\widehat{\mathcal{FF}}\big{]}(\xi): =\sqrt{\frac{\pi}{2\sigma^{2}}}\sum_{k=-\infty}^{\infty}\mathbf{x}_ {\mathbf{f}}[k]\,\mathrm{e}^{-j\pi k_{\sigma}^{2}/\sigma},\] \[\big{[}\mathcal{FF}\big{]}(\xi): =\frac{1}{\sqrt{2\pi}}\int_{t\in\mathbb{R}}\mathbf{f}(t)\,\mathrm{e} ^{-j\xi t}\;\;\mathrm{d}t= \begin{cases}\big{[}\widehat{\mathcal{FF}}\big{]} (\xi),&\text{if }-\sigma\leq\xi\leq\sigma,\\ 0,&\text{otherwise},\end{cases} \]  
for all \(\mathbf{f}\in\mathcal{B}_{\sigma}^{2}\), where \(\mathcal{FF}\) denotes the Fourier Transform of \(\mathbf{f}\) on the real line. Through the definition of the sinc-function,  
\[\operatorname*{sinc}(z):= \begin{cases}\frac{\sin(\pi z)}{\pi z}&\text{if }z\
eq 0, \\ 1&\text{if }z=0,,\end{cases} ,\quad z\in\mathbb{C},\]  
and the linearity of the Fourier Transform, the isometry provides [PERSON]'s original sampling theorem for the spaces \(\ell^{2}\) and \(\mathcal{B}_{\sigma}^{2}\), we have  
\[\lim_{N\to\infty}\int_{\mathbb{R}}\left|\mathbf{f}(t)-\sum_{k=-N}^{N}\mathbf{x}_{\mathbf{ f}}[k]\operatorname*{sinc}\big{(}t-(k\pi/\sigma)\big{)}\right|^{2}\,\mathrm{d}t=0.\]  
Since \(\mathcal{FF}\) is zero outside the interval \([-\sigma,\sigma]\), \(\mathbf{f}\) is called bandlimited with bandwidth \(\sigma\). The spaces \(\mathcal{B}_{\sigma}^{2}\), \(0<\sigma<\infty\), thus correspond to the traditional notion of bandlimited signals. Through the definition of exponential types (Point 2 of the requirements above), this notion is generalized to a significantly larger class of functions. For \(1<p<\infty\) arbitrary, the Plancherel-Polya theorem (Theorem 3, p. 152 in [11]) provides a nontrivial extension to the [PERSON]'s sampling theorem. **Theorem 1** (Plancherel-Polya). : _Let \(1<p<\infty\). For all sequences \(\mathbf{x}\in\ell^{p}\), \(1<p<\infty\), there exists a unique function \(\mathbf{f}\in\mathcal{B}^{p}_{\sigma}\), such that_  
\[\lim_{N\to\infty}\int_{\mathbb{R}}\left|\mathbf{f}(t)-\sum_{k=-N}^{N}\mathbf{x}[k] \operatorname{sinc}\big{(}t-(k\pi/\sigma)\big{)}\right|^{p}\,\mathrm{d}t=0\]  
_is satisfied. In particular, \(\mathbf{f}\) is the unique solution to the interpolation problem \(\mathbf{f}(k\pi/\sigma)=\mathbf{x}[k]\), \(k\in\mathbb{Z}\). Conversely, for all signals \(\mathbf{f}\in\mathcal{B}^{p}_{\sigma}\), \(1<p<\infty\), the sequence \((\mathbf{f}(k\pi/\sigma))_{k\in\mathbb{Z}}\) belongs to \(\ell^{p}\) and there exist constants \(C_{L}(p)>0\) and \(C_{R}(p)>0\), independent of \(\mathbf{f}\), such that_  
\[C_{L}(p)\sum_{k=-\infty}^{\infty}|f(k\pi/\sigma)|^{p}\leq\|\mathbf{f}\|_{p}^{p} \leq C_{R}(p)\sum_{k=-\infty}^{\infty}|\mathbf{f}(k\pi/\sigma)|^{p}\]  
_holds true._  
For the spaces \(\ell^{1}\), \(\mathcal{B}^{1}_{\sigma}\) and \(\ell^{\infty}_{0}\), \(\mathcal{B}^{\infty}_{0,\sigma}\), \(0<\sigma<\infty\), Theorem 1 does _not_ hold to its full extent. The (generalized) Plancherel-Polya Theorem, which we will subsequently refer to as generalized Shannon equivalence, provides the following: For all \(\mathbf{f}\in\mathcal{B}^{1}_{\pi}\), \(\mathbf{f}\in\mathcal{B}^{\infty}_{0,\pi}\), respectively, and \((f(k\pi/\sigma))_{k\in\mathbb{Z}}=\mathbf{x}_{f}\), we have \(\mathbf{f}\equiv 0\) if and only if we also have \(\mathbf{x}_{f}\equiv 0\).