A characterization of the convergence in variation for the generalized sampling series

TL;DR

This paper characterizes the convergence in variation of generalized sampling series using averaged kernels, linking it to the properties of absolutely continuous functions and providing examples with B-splines and classical kernels.

Contribution

It introduces a new characterization of convergence in variation for generalized sampling operators based on averaged kernels, connecting it to the derivative and sampling Kantorovich series.

Findings

Established a relation between the derivative of the sampling operator and the Kantorovich series.

Provided examples with B-splines, Fejer, and Bochner-Riesz kernels.

Proved a variation detracting property for the operators.

Abstract

In this paper, we study the convergence in variation for the generalized sampling operators based upon averaged-type kernels and we obtain a characterization of absolutely continuous functions. This result is proved exploiting a relation between the first derivative of the above operator acting on $f$ and the sampling Kantorovich series of f'. By such approach, also a variation detracting-type property is established. Finally, examples of averaged kernels are provided, such as the central B-splines of order $n$ (duration limited functions) or other families of kernels generated by the Fejer and the Bochner-Riesz kernels (bandlimited functions).