## 2 Sampling Kantorovich (SK) Mathematical Model  
To weaken the assumptions connected with the generalized sampling theorem and to extend the convergence results to the class of functions belonging to \(L^{p}\) spaces, the Sampling Kantorovich (SK) operators have been introduced in [15] as more suitable in this setting.  
In the one dimensional case, let \(\Pi=(t_{k})_{k\in\mathbb{Z}}\), a sequence of real numbers such that:  
* \(-\infty<t_{k}<t_{k+1}<+\infty\) ;
* \(\lim\limits_{k\to\pm\infty}t_{k}=\pm\infty\) ;
* there exists \(\Delta,\delta>0\) such that \(\delta\leq t_{k+1}-t_{k}:=\Delta_{k}\leq\Delta\), for every \(k\in\mathbb{Z}\). We call \(\Pi\) a subdivision of \(\mathbb{R}\) and it is fixed throughout the paper. A function \(\chi:\mathbb{R}\to\mathbb{R}\) is called a _kernel_ for the SK operator if:
* \(\chi\in L^{1}(\mathbb{R})\) and it is bounded in a neighborhood of the origin;
* \(\sum_{k\in\mathbb{Z}}\chi(u-t_{k})=1\), for each \(u\in\mathbb{R}\);* there exists \(\beta>0\), such that \[\begin{array}{l} m_{\beta,\mathrm{I}1}(\chi):=\sup_{u\in\mathbb{R }}\sum_{k\in\mathbb{Z}}|\chi(u-t_{k})||u-t_{k}|^{\beta}<+\infty,\,\text{i.e., the discrete absolute moments of order $\beta$ are finite.}\end{array}\]  
**Definition 1**.: _Given a kernel \(\chi\), the family_  
\[(K_{w}f)(x):=\sum_{k\in\mathbb{Z}}\chi(ux-t_{k})\left[\tfrac{\mathbb{E}}{ \Delta_{k}}f_{\frac{k}{\mathbb{R}}}^{\frac{k+1}{\alpha}}\,f(u)\;du\right]\]  
_for \(x\in\mathbb{R}\), \(w>0\) and where \(f:\mathbb{R}\to\mathbb{R}\) is a locally integrable function such that the above series converges for each \(x\in\mathbb{R}\), defines the sampling Kantorovich operators._  
It is possible to extend the same family of operators in a multidimensional setting ([22]). The Definition 1 is given in the general context of non-uniform sampling schemes, but it includes as a particular case the equi-spaced sampling, often used in the applications.  
Examples of well-known kernels satisfying the above assumptions and used in the implementation of the SK operators, in one-dimensional as well in multidimensional settings, are given in [10; 11; 13; 23; 24].  
SK are approximation operators, able to reconstruct pointwise, continuous, and bounded functions, and uniformly, functions which are uniformly continuous and bounded, as \(w\to+\infty\), (see, e.g., Refs. [15; 22; 25; 26]).  
Convergence theorems stand (see, e.g., Ref. [22]).  
**Theorem 2**.: _Let \(\chi\) be a kernel for the operators defined in Definition 1 and let \(f:\mathbb{R}^{n}\to\mathbb{R}\) be a continuous and bounded function. Then, for every \(\underline{x}\in\mathbb{R}^{n}\),_  
\[\lim_{w\to+\infty}(K_{w}f)(\underline{x})\;=\;f(\underline{x}).\]  
_Moreover, if \(f\in C(\mathbb{R}^{n})\), then_  
\[\lim_{w\to+\infty}\|K_{w}f-f\|_{\infty}\;=\;0,\]  
_where \(\|\cdot\|_{\infty}\) denotes the usual sup-norm._  
Moreover, the operators \(K_{w}\) are suitable also to reconstruct not necessarily continuous signals, as happens, e.g., in \(L^{p}\)-setting. For this aim, it is possible to formulate the following theorem.  
**Theorem 3**.: _Let \(\chi\) be a kernel for the operators defined in Definition 1. For every \(f\in L^{p}(\mathbb{R}^{n})\), \(1\leq p<+\infty\), there holds_  
\[\lim_{w\to+\infty}\|K_{w}f-f\|_{p}\;=\;0,\]  
_where \(\|\cdot\|_{p}\) denotes the usual \(L^{p}\)-norm._  
For the above \(SK\) operators, results concerning rate of approximation, saturation order, and inverse theorems, can be found in [18; 27; 28; 29]. Moreover, the above operators have been implemented for image reconstruction and enhancement and used for various applications (see, e.g., Refs. [24; 30]), exploiting their low pass filtering properties (see, e.g., Ref. [31]).