## 1 Introduction
The above operators are bounded linear operators mapping \(L^{\infty}(\mathbb{R})\) into \(L^{\infty}(\mathbb{R})\) and \(C(\mathbb{R})\) into itself, with respect to the \(L^{\infty}(\mathbb{R})\) and \(C(\mathbb{R})\) norms, respectively, (here \(L^{\infty}(\mathbb{R})\) denotes the space of essentially bounded functions, while \(C(\mathbb{R})\) is the space of bounded and uniformly continuous functions on \(\mathbb{R}\)). The operator norm is \(\|S_{w}^{\varphi}\|_{C(\mathbb{R}),C(\mathbb{R})}|=\sup_{u\in\mathbb{R}}\sum _{k\in\mathbb{Z}}|\varphi(u-k)|\), where \(m_{0}(\varphi)=\sup_{u\in\mathbb{R}}\sum_{k\in\mathbb{Z}}|\varphi(u-k)|\) defines the absolute discrete moment of order zero of the function \(\varphi\). Moreover, it is possible to prove (see, e.g., Ref. [9]) that \((S_{w}^{\varphi}f)(x)\) converges to \(f(x)\) as \(w\) tends to \(+\infty\), when the kernel \(\varphi\) satisfies the following condition:  
\[\sum_{k\in\mathbb{Z}}\varphi(x-k)\ =\ 1,\]  
for every \(x\in\mathbb{R}\), pointwise in every point \(x\) of continuity for \(f\) bounded and uniformly with respect to \(x\in\mathbb{R}\), when \(f\) is bounded and uniformly continuous. The generalized sampling formulation has, compared to the classical WKS sampling theorem, the advantages to require only a finite number of samples, that in addition, can be taken only in the past with respect to the point to reconstruct (this last result being valid assuming \(\varphi\in C_{c}(\mathbb{R}^{+})\)). As previously mentioned, the weakening of the hypotheses has its downside, represented by the introduction of a reconstruction error. In the setting of the generalized operators, controlling this error is important to quantify the goodness of the reconstruction: for these reasons, besides results of convergence, also results about the order of approximation, saturation order, and inverse results, have been given (see, e.g., Ref. [11]). For the extension of the theory to a multidimensional setting and to other functional spaces, see ([12; 13; 14]). Note that thanks to the generalized formulation, the problem concerning the not causality condition of the reconstruction process is avoided (for \(\varphi\in C_{c}(\mathbb{R}^{+})\)). From a practical point of view, the generalized formulation maintains a pointwise mathematical model, in opposition to the not-pointwise nature of the samples in the measurement process. Indeed, a certain time is required to perform an acquisition, but neither the WKS nor the generalized formulation models take into account of this aspect. As we see in Section 2, the Sampling Kantorovich mathematical model will overpass this limitation. Although the theory of reconstruction using these operators has solved the drawbacks connected with the classical sampling formulation, they are not very suitable to be studied in the context of not continuous functions/signals. In fact, it is not difficult to show (see the counterexample in [15]) that they do not map \(L^{p}(\mathbb{R})\) into itself, their expression showing how strongly they are related to the function values on single nodes \(\frac{k}{w},k\in\mathbb{Z},w>0\). A more realistic way to model the effective sampling process is to introduce, in place of the pointwise samples \(f(\frac{k}{w})\), an averaged version in a neighborhood of the nodes. From a theoretical point of view, to allow the reconstruction of a broader class of signals (therefore not necessarily restricted to \(C(R)\) or to \(C^{0}(R)\)-space of continuous functions), in [15] the Sampling Kantorovich (SK) operators have been introduced and studied. On the other hand, from an application point of view, the new operators mathematically model, concretely, the sampling process. Thanks to this model, we are able to define a sampling paradigm that we call Sampling Kantorovich by Difference (SKD), that is applied to a super-resolution-type problem. The goal of this work is to analyze the SK model and its behavior when applied to a sampling type problem, related, e.g., to a super resolution problem by multiple acquisitions.