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- Author or Editor: R. Patterson x
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Abstract
In this paper definitions for “bounded variation”, “subsequences”, “Pringsheim limit points”, and “stretchings” of a double sequence are presented. Using these definitions and the notion of regularity for four dimensional matrices, the following two questions will be answered. First, if there exists a four dimensional regular matrix A such that Ay = Σ k,l=1,1 ∞∞ a m,n,k,l y k,l is of bounded variation (BV) for every subsequence y of x, does it necessarily follow that x ∈ BV? Second, if there exists a four dimensional regular matrix A such that Ay ∈ BV for all stretchings y of x, does it necessarily follow that x ∈ BV? Also some natural implications and variations of the two Tauberian questions above will be presented.
Abstract
The concepts of subsequence and rearrangement of double sequence are used to present multidimensional analogues of the following core questions. If x is a bounded real sequence and A is a matrix summability method, under what conditions does there exist y, a subsequence (rearrangement) of x such that each number t in the core of x is a limit point of Ay?
Abstract
In 1930 Knopp presented the following matrix characterization for the core of ordinary sequences. If A is a nonnegative regular matrix then the core of [Ax] is contained in the core of [x], provided that [Ax] exists. Patterson in 1999 extended Knopp’s results to double sequences via four dimensional matrices. In a manner similar to the Knopp’s and Patterson’s results we present multidimensional extensions of Bustoz’s singular dimensional Gibbs phenomenon results. These results include a notion of what it means for a four dimensional matrix transformation to induce the double Gibbs phenomenon in s. In addition, necessary and sufficient conditions for a four dimensional matrix to induce the double Gibbs phenomenon is also presented.
