## 2 Materials and Methods  
In the following, we provide a concise introduction to the mathematics of sampling and interpolation, which are primarily based on the theory of Banach spaces and linear operators. To this end, we introduce the Banach spaces \(\ell_{0}^{\infty}\), \(\ell^{p}\), \(\mathcal{B}_{0,\sigma}^{\infty}\), and \(\mathcal{B}_{\sigma}^{p}\), \(1\leq p<\infty\), \(0<\sigma<\infty\). Commonly, \(\mathcal{B}_{0,\sigma}^{\infty}\), and \(\mathcal{B}_{\sigma}^{p}\) are referred to as Bernstein spaces. For a comprehensive introduction, we refer the reader to [11; 12]. By \(\ell_{0}^{\infty}\), we denote the set of all complex-valued sequences indexed by \(\mathbb{Z}\) that vanish at infinity. That is, we have  
\[\lim_{k\to\infty}\boldsymbol{x}[k]=\lim_{k\to-\infty}\boldsymbol{x}[k]=0\]  
Figure 2: Generalized Plancherel–Polya Theorem (c.f. Section 2). By evaluating a bandlimited, continuous-time signal (**left**) at all integer multiples of a suitable sampling interval \(T\), we obtain the sequence of samples (**right**) corresponding to the function under consideration. We can restore the function through an (infinite) interpolation series, where each term is associated with one of the sample sequence’s components. In particular, each term consists of a time-shifted interpolation function multiplied by the associated sample. This way, the bandlimited function and the sequence of sampling values uniquely determine each other. for all \(\mathbf{x}=(\mathbf{x}[k])_{k\in\mathbb{Z}}\in\ell_{0}^{\infty}\). Equipped with the uniform norm \(\|\mathbf{x}\|_{\infty}:=\sup_{k\in\mathbb{Z}}|\mathbf{x}[k]|\) the set \(\ell_{0}^{\infty}\) becomes a Banach space. Further, by \(\ell^{p}\), \(1\leq p<\infty\), we denote the Banach space of \(p\)th-power-summable sequences with the \(p\)-norm  
\[\|\mathbf{x}\|_{p}:=\left(\sum_{k=-\infty}^{\infty}|\mathbf{x}[k]|^{p}\right)^{1/p}.\]  
A function \(\mathbf{f}:\mathbb{C}\to\mathbb{C},z\mapsto\mathbf{f}(z)\) is called entire if it is well-defined and holomorphic on all of \(\mathbb{C}\). For entire functions that are (essentially) bounded on the real line, we define the essential-supremum norm \(\|\mathbf{f}\|_{\infty}:=\operatorname*{ess\,sup}_{t\in\mathbb{R}}|\mathbf{f}(t)|\). The space \(\mathcal{B}_{0,\sigma}^{\infty}\), \(0<\sigma<\infty\), consists of all entire functions \(\mathbf{f}\) that satisfy the conditions:  
1. \(\mathbf{f}\) is (essentially) bounded on the real line;
2. \(\mathbf{f}\) of exponential type (c.f. [11], Lecture 1, p. 3 ff) at most \(\sigma\);
3. \(\mathbf{f}(t)\) vanishes for \(t\to\pm\infty\) on the real line. Equipped with the essential supremum norm, the space \(\mathcal{B}_{0,\sigma}^{\infty}\) becomes a Banach space. Further, the Banach spaces \(\mathcal{B}_{\sigma}^{p}\), \(1\leq p<\infty\), consists of all functions in \(\mathcal{B}_{0,\sigma}^{\infty}\) that are \(p\)th-power integrable on the real line, equipped with the \(p\)-norm \[\|\mathbf{f}\|_{p}:=\left(\int_{t\in\mathbb{R}}|\mathbf{f}(t)|^{p}\;\mathrm{d}t\right)^ {1/p}.\]  
Pure mathematics studies all of the spaces \(\ell_{0}^{\infty}\), \(\ell^{p}\), \(\mathcal{B}_{0,\sigma}^{\infty}\), and \(\mathcal{B}_{\sigma}^{p}\), \(1\leq p<\infty\), \(0<\sigma<\infty\). In contrast, only the spaces \(\ell^{1}\), \(\ell^{2}\), \(\ell_{0}^{\infty}\) and \(\mathcal{B}_{\sigma}^{1}\), \(\mathcal{B}_{\sigma}^{2}\), \(\mathcal{B}_{0,\sigma}^{\infty}\) occur frequently throughout signal processing and communications engineering, the arguably most \"well-known\" ones being \(\ell^{2}\) and \(\mathcal{B}_{\sigma}^{2}\). They consist of discrete- and, respectively, bandlimited continuous-time signals with finite energy and form the mathematical basis for the seminal results in SSI, established before the relevant theory was extended to general Bernstein spaces.