Tauberian conditions connecting Cesaro and Riesz discrete means methods

Abstract

Everywhere below, if not stated otherwise, we assume that {an} n=0 is a sequence of real numbers; ∑ an is the series corresponding to it (if summation limits are not specified, we assume summation from 0 to +∞). Let Ω and Λ be summation methods for number series. The summability of a series ∑ an to a number S by the method Ω is briefly denoted as ∑ an = S(Ω). We say that the method Λ is included into the method Ω (Λ ⊂ Ω) if the equality ∑ an = S(Λ) implies ∑ an = S(Ω). We say that the methods Ω and Λ are equivalent (Ω ∼ Λ) if both the inclusions Λ ⊂ Ω and Ω ⊂ Λ are valid. We say that the method Ω is stronger than the method Λ if Λ ⊂ Ω, but the methods Ω and Λ are not equivalent. For Ω and Λ we consider the summation method of Cesaro (C,α) and methods of summation by Riesz means denoted by (Rd, α), where α > −1. Definitions and basic properties of Cesaro methods can be found in [1, § 5.4–5.7]; Riesz methods were described in [2, 3]. Present here the corresponding definitions in the form convenient for further use. Definition 1. A series ∑ an is said to be summable by the Cesaro method of order α to a number S if