## 1 Introduction  
Every aspect of our daily life is strongly connected with the quality and quantity of information we can perceive and elaborate. Information is contained in signals, transmitted in different forms and via different physical mediums. To measure these signals and extract useful information from them, specific techniques have been developed. The main problem of reconstructing signals, starting from the knowledge of their samples, has been tackled and mathematically solved by the following classic Whittaker-Kotelnikov-Shannon (WKS) theorem. **Theorem 1** (WKS Sampling Theorem [1]). : _Let \(f\in L^{2}(\mathbb{R})\) such that \(supp\ \widehat{f}\subseteq[-\pi w,\pi w]\), \(w>0\), where \(\widehat{f}\) denotes the Fourier transform of the function \(f\). Then, \(f\) can be reconstructed on the whole real axis by the following formula_  
\[\sum_{k\in\mathbb{Z}}f\bigg{(}\frac{k}{w}\bigg{)}sinc(wt-k)\ =\ f(t),\ \ \ \ \ \ \ \ t\in\mathbb{R}.\]  
The WKS sampling theorem states the sufficient conditions for the exact reconstruction of a function \(f\in L^{2}(\mathbb{R})\), which is band-limited, by means of the above interpolation formula given its samples values \(f(\frac{k}{w})\), \(k\in\mathbb{Z},w>0\) calculated in the, so-called nodes \(\frac{k}{w}\). For an interesting discussion about the above formula, see, e.g., Ref. [2]. From an application point of view, the WKS theorem states that, given an infinite countable set of samples \(f\Big{(}\frac{k}{w}\Big{)}\), it is possible to reconstruct the original signal without any errors if the Nyquist condition \(v_{s}\geq 2v_{max}\) is satisfied ([3,4]), where \(v_{s}\) is the sampling frequency and \(v_{max}\) is the maximum frequency of the sampled signal. Although the WKS theorem had a very strong impact as in the mathematical literature as in the applications, its hypotheses hide both theoretical as practical limitations. From the theoretical point of view, to require that \(f\in L^{2}(\mathbb{R})\) and \(supp\,\widehat{f}\subseteq[-\pi w,\pi w]\), \(w>0\), means that \(f\) is the restriction to the real axis of an entire function of exponential type (see the Paley-Wiener theorem [5]). Moreover, from the application point of view, the above interpolation formula requires the knowledge of the samples values \(f\Big{(}\frac{k}{w}\Big{)}\) over whole \(\mathbb{R}\), being \(k\in\mathbb{Z}\), i.e., the sampling of the signal on infinite time nodes: consequently, the evaluation of the signal should be made both in the past of a fixed instant \(t\in\mathbb{R}\) as in the future, which means making a prediction (causality condition is not respected). Furthermore, in concrete application problems, signals are duration-limited (i.e., with compact support) and this is not compatible with the requested band limitation, as stated by the Heisenberg uncertainty principle. Moreover, in applications, the value of \(\
u_{max}\) is generally fixed by the degree of precision requested in the measurement and not by the signal itself: in fact, it is not possible to precisely state a priori what is the maximum frequency of an unknown function. For all these reasons, the hypotheses of the WKS sampling theorem need to be weakened. Relaxing the assumptions determines the loss of an exact reconstruction and consequently introduces an approximation error. A class of operators, called generalized sampling operators, has been formulated with the aim to solve the mentioned critical issues connected with the WKS theorem (see, e.g., Refs. [6; 7; 8; 9]). For a function \(\varphi:\mathbb{R}\to\mathbb{R}\) with \(\varphi\in C_{c}(\mathbb{R})\) (where \(C_{c}(\mathbb{R})\) is the space of continuous functions with compact support on \(\mathbb{R}\)) and a bounded function \(f:\mathbb{R}\to\mathbb{R}\), the generalized sampling operators are defined as  
\[(S_{w}^{\varphi}f)(x):=\sum_{k\in\mathbb{Z}}f\Big{(}\frac{k}{w}\Big{)}\varphi( wx-k),\,x\in\mathbb{R},\,w>0\,,\]  
for every \(x\in\mathbb{R}\), \(\varphi\) being a suitable kernel function (see, e.g., Refs. [9; 10]).