A bounded consistency theorem for strong summabilities

The study of R-type summability methods is continued in this paper by showing that two such methods are identical on the bounded portion of the strong summability field associated with the methods. It is shown that this “bounded consistency” applies for many non-matrix methods as well as for regular matrix methods.

Generalized Nörlund means and consistency theorems

It was recently shown in (1), corollary 1, that if (I) = (N, q) then I and (N, q) were consistent. By I we denote the unit infinite matrix, and (N, q) denotes a general complex Nörlund mean. The summability field (A) of any infinite matrix A = (ank) is defined aswhere c is the space of convergent sequences. The summability field of (N, q) is, for simplicity, written as (N, q) instead of ((N, q)).

Consistency and Nörlund means

Let A = (ank) be an infinite matrix of complex numbers; suppose I is the unit matrix and c is the space of convergent sequences. By (A) we denote the summability field of A, i.e. (A) = {x = (xk): Ax ∈ c}.

Uniform summability of power series

It is shown that for the Cesàro means (C, α) with α > - 1, and for a certain class of more general Nörlund means, summability of the series σan implies uniform summability of the series σan zn in a Stolz angle at z = 1.If B is a normal matrix and (B) denotes the series summability field with the usual Banach space topology, then the vectors {ek} (ek = {0,0,..., 1,0,...}) are said to form a Toplitz basis for (B) relative to a method H if H — Σakek = a for each a = {ak}ε(B). It is shown for example that the above relation holds for B = (C,α), α> − 1 , and H = Abel method; also for B = (C,α) and H = (C,β) with 0 ≤ α ≤ β.Applications are made to theorems on summability factors.

Approximation in Bounded Summability Fields

This paper deals with several related properties of bounded summability fields of regular, real matrices. For a matrix A = (ank) and a sequence x = {xn}, we write formallyWe denote by m the space of bounded real sequences, and by A* the bounded summability fieldof A. The strong summability field of A is the set