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

In this paper, we study the convergence in variation for the generalized sampling series, a family of operators that are well-known in Approximation Theory and that play a relevant role in Signal Processing. Results about variation of the generalized sampling series were previously studied in some particular cases, for example in case of band-limited kernels, but the topic was never faced in a general setting. In particular, in case of averaged-type kernels, we obtain a complete characterization of the absolutely continuous functions in terms of convergence in variation by means of such family of discrete operators. This result is proved exploiting a relation between the first derivative of the above operator acting on f and the sampling Kantorovich series of the derivative of f: such result is similar to the well-known relation between the Bernstein polynomials and their Kantorovich version. By means of such approach, also a variation detracting-type property is established. Finally, examples of averaged kernels are presented, hence providing several families of classical kernels to which our results can be applied: among them, the central B-splines of order n (duration limited functions) or other families of kernels generated by the Fejér and the Bochner–Riesz kernels (bandlimited functions).