Approximation of differentiable and not differentiable signals by the first derivative of sampling Kantorovich operators
Approximation by sampling type operators is one of the topics that have been deeply studied in recent years. Methods and techniques from Approximation Theory, Fourier and Harmonic Analysis have been used in these studies. The origin of this research field resides in the well-known Whittaker-Kotelnikov-Shannon sampling theorem. Generalized sampling operators have been considered in order to extend the classical theory. The so-called sampling Kantorovich operators have been introduced in [\textit{C. Bardaro} et al., Sampl. Theory Signal Image Process. 6, No. 1, 29--52 (2007; Zbl 1156.41307)] with the aim to provide a family of sampling-type operators useful in order to reconstruct signals which are not necessarily continuous. The latter necessity is motivated by the applications, where in real world cases signals are not quite regular. In this paper the behavior of the first derivative of the sampling Kantorovich operators is studied, and both differentiable and not differentiable signals are taken into account. In particular, the authors prove that the family of the first derivatives of the above operators converges pointwise at points of differentiability of f, and uniformly to the first derivative of a given continuously differentiable signal. Moreover, the convergence is analysed also in presence of points in which a signal is not differentiable. In this case, under suitable assumptions, the authors prove 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, the extension of the above results to the case of signals with higher order derivatives is discussed. At the end of the paper examples of kernel functions and graphical representations are provided. Furthermore, numerical examples in the cases of functions for which the left and right derivatives are infinite or they do not exist, are given.
approximation at not differentiable points
scientific article; zbMATH DE number 7461182
scientific article