A new convergence method based on local block averages

In this paper, a new mode of convergence for sequences, called Drift-Controlled Local Ces ro convergence (DLC-convergence for short), is introduced.This notion requires local Ces ro averages taken over blocks whose lengths tend to infinity, but remain small compared to the global index, and whose drift between successive blocks is bounded.Basic properties of this convergence are established, comparisons with classical and Ces ro convergence are provided, counterexamples separating these notions are given, stability under Toeplitz operators is proved, and a Tauberian-type theorem is presented.