Richard F. Patterson

Analogues of some fundamental theorems of summability theory

In 1911, Steinhaus presented the following theorem: if A is a regular matrix then there exists a sequence of 0s and 1s which is not A-summable. In 1943, R. C. Buck characterized convergent sequences as follows: a sequence x is convergent if and only if there exists a regular matrix A which sums every subsequence of x. In this paper, defini- tions for subsequences of a double sequence and Pringsheim limit points of a double sequence are introduced. In addition, multidimensional analogues of Steinhaus and Bucks theorems are proved.

Double sequence core theorems

In 1900, Pringsheim gave a definition of the convergence of double sequences. In this paper, that notion is extended by presenting definitions for the limit inferior and limit superior of double sequences. Also the core of a double sequence is defined. By using these definitions and the notion of regularity for 4-dimensional matrices, extensions, and variations of the Knopp Core theorem are proved.

On asymptotically statistical equivalent sequences

Abstrac t . This paper presents the following definition which is a natural combination of the definition for Asymptotically equivalent and Statistically limit. Two nonnegative sequences [x] and [y] are said to be asymptotically statistical equivalents of multiple L provided that for every e > 0, limn ¿ { t h e number of k < n : — L > f } = 0 (denoted ^L by a: ~ y), and simply asymptotically statistical equivalent if L = 1. In addition, there are also statistical analogs of theorems of Poyvanents in [5].

A theorem on entire four dimensional summability methods

This paper introduce the following concept of double entire double sequences. Let x={xk,l} be a double sequence of complex numbers. Then the double sequence {x} is entire provided [emailxa0protected]?k,l=0,0~,~xk,lp^kq^l<~for every positive integers p and q. Using this definition I will also introduce the notion Four dimensional entire summability method. The author uses the notions to prove directly necessary and sufficient condition which ensure that a four dimensional transformation is entire.

A category theorem for double sequences

Abstract The goal of this paper is to present the following multidimensional category theorem for double sequences via four-dimensional matrix transformations. The set of second category double subsequences of the double sequence ( s k , l ) is summable by a four-dimensional RH-regular matrix A = ( a m , n , k , l ) if and only if ( s k , l ) is P -convergent. This theorem is established using multidimensional sliding hump constructions for bounded and unbounded doubles sequences.

Characterization for the limit points of stretched double sequences

This paper investigates the effect of four dimensional matrix transformation on new classes of double sequences. Subsequences and stretchings of a double sequence are defined, and these definitions are used to present a four dimensional analogue of D. Dawsons Copy theorem for stretchings of a double sequence. In addition, the multidimensional analogue of D. Dawsons Copy theorem is used to characterize convergent double sequences using subsequences and stretchings.

Rate preservation of double sequences under l-l type transformation

Following the concepts of divergent rate preservation for ordinary sequences, we present a notion of rates preservation of divergent double sequences under l-l type transformations. Definitions for Pringsheim limit inferior and superior are also presented. These definitions and the notion of asymptotically equivalent double sequences, are used to present necessary and sufficient conditions on the entries of a four-dimensional matrix such that, the rate of divergence is preserved for a given double sequences under l-l type mapping where l=:{xk,l:∑k,l=1,1∞,∞|xk,l|<∞}.

λ-Rearrangements Characterization of Pringsheim Limit Points

Sufficient conditions are given to assure that a four-dimensional matrix A will have the property that any double sequence x with finite P-limit point has- a λ-rearrangement z such that each finite P-limit point of x is a P-limit point of Az.

ANALOGUES OF SOME TAUBERIAN THEOREMS FOR STRETCHINGS

We investigate the effect of four-dimensional matrix transformation on new classes of double sequences. Stretchings of a double sequence is defined, and this defi- nition is used to present a four-dimensional analogue of D. Dawsons copy theorem for stretching of a double sequence. In addition, the multidimensional analogue of D. Dawsons copy theorem is used to characterize convergent double sequences using stretchings.