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

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).