Inclusion relations among methods of four-dimensional summability compounded from given four-dimensional methods
Richard F. Patterson

Abstract

The goals of this paper include the introduction of a new four-dimensional summability method construction by compounding a single four-dimensional method. The examination of this method begins with the characterization of its RH-regularity properties. In addition, the following inclusion and consistent theorems will be presented.

If {α′ m } , {β′ n } , {α _m }, {β _n } are sequences such that {α _m } and {β _n } are monotone increasing with α′ m ≥αm and β′ n ≥βn for all sufficiently large m and n and if the transformations B(α′ m ,β′ n )   and B(α _m ,β _n ) are factorable and RH-regular then B(α′ m ,β′ n )   includes B(α _m ,β _n ).

The RH-regular matrix transformations of the form B(r _m ,s _n ) for which r₁≤r₂≤r₃≤⋯ and s₁≤s₂≤s₃≤⋯ constitute a double sequence of consistent family. Other implications and variations will also be presented.