Approximation of differentiable and not differentiable signals by the first derivative of sampling Kantorovich operators

In this paper the behavior of the first derivative of the so-called sampling Kantorovich operators has been studied, when both differentiable and not differentiable signals are taken into account. In particular, we proved that the family of the first derivatives of the above operators converges pointwise at the point of differentiability of f, and uniformly to the first derivative of a given C1 signal. Moreover, the convergence has been analysed also in presence of points in which a signal is not differentiable. In the latter case, under suitable assumptions, it can be proved the pointwise convergence of the first derivative of the above operators to a certain combination of the left and right first derivatives (whenever they exist and are both finite) of the signals under consideration. Finally, also the extension of the above results to the case of signals with high order derivatives has been discussed. At the end of the paper, examples of kernel functions and graphical representations are provided. Further, also numerical examples in the cases of functions for which the left and right derivatives are ±∞ or they do not exist in R˜, have been given.