## 3 Sampling Kantorovich by Difference
We define a sensor by using an integral model, such that, given an input, the sensor returns to the output the mean value of the given input. Moreover, in our case we assume to be in an ideal situation, i.e., without any distortion or non-linearity. For this reason, given the aim of the work is the theoretical investigation of the SK model, we will use simulated data in Section 4. Between the two measurements, the sensor is shifted by \(\frac{d}{2}\). We also assume the measured values of \(f\) depending on \(d\), in a similar way to what happens with light intensity in photographic cameras. The process is schematized in Figure 2: the first measurement is performed with the sensor in position B (upper row); then the sensor is shifted to the right by \(\varepsilon=\frac{d}{2}\) and another measurement is performed in position A (second row). These two measurements provide the values of \(M_{B}\) and \(M_{A}\), respectively. The goal is to achieve a doubled resolution with respect to the size \(d\). Then,  
\[M_{1}\Big{(}\bigg{[}\frac{t_{k_{A}}}{w},\frac{t_{k_{B}+1}}{w}\bigg{]}\Big{)}= 2M_{B}\Big{(}\frac{t_{k_{B}}}{w}\Big{)}-M_{0}\Big{(}\bigg{[}\frac{t_{k_{B}}}{w },\frac{t_{k_{A}}}{w}\bigg{]}\Big{)},\]  
\[M_{2}\bigg{(}\bigg{[}\frac{t_{k_{B}+1}}{w},\frac{t_{k_{A}+1}}{w}\bigg{]}\bigg{)} =2M_{A}\bigg{(}\frac{t_{k_{A}}}{w}\bigg{)}-\bigg{[}2M_{B}\bigg{(}\frac{t_{k_{B} }}{w}\bigg{)}-M_{0}\bigg{(}\bigg{[}\frac{t_{k_{B}}}{w},\frac{t_{k_{A}}}{w} \bigg{]}\bigg{)}\bigg{]}.\]  
Same equations can be achieved iterating, obtaining, for \(p\in\mathbb{N}\),  
\[t_{k_{B}+h}=\frac{t_{k_{A}+h}+t_{k_{A}+h-1}}{2},\]0  
\[t_{k_{B}+h}=\frac{t_{k_{A}+h}+t_{k_{A}+h-1}}{2},\]1  
\[t_{k_{B}+h}=\frac{t_{k_{A}+h}+t_{k_{A}+h-1}}{2},\]2  
\[t_{k_{B}+h}=\frac{t_{k_{A}+h}+t_{k_{A}+h-1}}{2},\]3  
with \(M_{0}\) as above defined, i.e., the applicability of the method is represented by the knowledge of \(M_{0}\). Indeed, in practice, \(M_{0}\) should be measured by a sensor of size \(d\). This implies that \(M_{0}\) is known if, e.g., \(f\) is constant on the interval \([\frac{t_{k_{B}}}{w},\frac{t_{k_{B}+1}}{w}]\) or the mean of \(f\) in \([\frac{t_{k_{B}}}{w},\frac{t_{k_{A}}}{w}]\) and in \([\frac{t_{k_{A}}}{w},\frac{t_{k_{B}+1}}{w}]\) is equal to \(M_{0}\). Therefore, from now on, we will define the following \"far field\" condition. Definition 2: The signal \(f\) satisfies the far field condition on the interval \(I\subset\mathbb{R}\), with \(I=I_{1}\cup I_{2}\), \(I_{1}^{\circ}\cap I_{2}^{\circ}=\emptyset\), if either of the two statements below hold true:  
* \(f(x)=C,\,C\in\mathbb{R},\,\,\,\,\,\text{for every \[t_{k_{B}+h}=\frac{t_{k_{A}+h}+t_{k_{A}+h-1}}{2},\]4}\),  
\[t_{k_{B}+h}=\frac{t_{k_{A}+h}+t_{k_{A}+h-1}}{2},\]5 Sensor shifting in the measurements setup: from the starting position B, the subsequent measurement is achieved moving on the right by \(\frac{d}{2}\) (position A). \(\bullet\)\(\frac{1}{|f_{1}|}\int_{I_{1}}f(x)dx=\frac{1}{|f_{2}|}\int_{I_{2}}f(x)dx\),  
_where by \(|\cdot|\) we denote the measure of the considered interval._  
Note that the above far field condition consists in knowing the numeric value for \(M_{0}\), from which the calculation of \(M_{A}\) and \(M_{B}\) follows. From the application point of view the far field conditions can be always achieved, using suitable technical solutions. Note that the above condition can be formulated in multidimensional settings, extending the reasoning in more variables. Moreover, we highlight that in the present setting, the partitions are not necessarily constituted by integers numbers, i.e., in general \(t_{k}\
eq k,k\in\mathbb{Z}\), as could happen in practical situations. **Remark 2**. : _We remark that, since the two sequences \(\Pi_{A}=(t_{k_{A}})_{k_{A}\in\mathbb{Z}}\), \(\Pi_{B}=(t_{k_{B}})_{k_{B}\in\mathbb{Z}}\), are equally spaced each other, we could have used a single partition and achieve a unique expression for \(M_{p}\). We preferred to formalize the model by two distinct partitions to clearly distinguish the starting sequence and the one obtained by shifting the sensor.