A characterization of the absolute continuity in terms of convergence in variation for the sampling Kantorovich operators

In the last forty years, the study of sampling-type operators represented a widely investigated topic, in view of their deep relations with Approximation Theory, Signal and Image Processing, from both theoretical and applicative aspects. In order to study a family of sampling-type operators that can be used also to approximate not necessarily continuous signal, the family of sampling Kantorovich operators was introduced. The idea, with respect to the classical generalized sampling operators, is similar to that one used for the well-known Bernstein polynomials extended in the L^p-setting by the Bernstein-Kantorovich polynomials, namely to replace the sampling values are replaced by mean values of the function. In this paper we study the problem of the convergence in variation for the sampling Kantorovich series. Due to the form of the operators, the problem is quite delicate and a direct approach, that is often used working with the other classes of integral operators, here cannot be followed. To reach our goal, we use a relation between the sampling Kantorovich series and the generalized sampling series, obtaining a characterization of the space of the absolutely continuous functions in terms of the convergence in variation by means of the sampling Kantorovich operators in case of averaged-type kernels. Moreover, in case of classical band-limited kernels, not necessarily of averaged form, we prove convergence in variation in case of functions belonging to a Bernstein space. Several examples of kernels to which the results can be applied are also provided.