Berezin symbols and Borel summability

Abstract

We prove in terms of so-called Berezin symbols some theorems for Borel summability method for sequences and series of complex numbers. Namely, we characterize the Borel convergent sequences and series; prove regularity of Borel summability method, and prove a new Tauberian type theorem for Borel summability.

Keywords: Borel summabiliy, Berezin symbol, diagonal operator, compact operator, (e)-convergence, Tauberian type theorem, Fock space