Matrix transforms into the speed-Maddox spaces over ultrametric fields

In the present paper, K denotes a complete, non-trivially valued, ultrametric (or non-archimedean) field. Entries of sequences, infinite series and infinite matrices are in K. Following Kangro [2, 3, 4], we introduce the concept of boundedness with speed λ or λ-boundedness. We then obtain a characterization of the matrix class (mλ , mµ ), where mλ denotes the set of all λ-bounded sequences in K. We conclude the paper with a remark about the matrix class (c λ , mµ ), where c λ denotes the set of all λ-convergent sequences in K.

In this paper, K denotes a complete, non-trivially valued, non- archimedean field. The entries of sequences, series and infinite matrices are in K. In the present paper, we prove the Knopp's core theorem for double sequences in K and also the necessary and sufficient conditions for the core t be invariant under the four dimensional matrix transformation in such fields.

In this paper, K denotes a complete, non-trivially valued, non-archimedean field. The entries of sequences, series and infinite matrices are in K. In the present paper, we prove the Silvermann-Toeplitz theorem for double sequences and series in K and apply it to Norlund means for double sequences and series in K.

In this paper, entries of sequences, infinite series and infinite matrices are real or complex numbers. The present paper is a continuation of [8], where we established some properties of the matrix class \((\ell_\alpha, \ell_\alpha)\), \(0 < \alpha \leq 1\). In this paper, we also record some properties of the class \((\ell_\alpha, c)\) and the sequence space \(\ell_\alpha\), \(0 < \alpha \leq 1\).

Throughout the present paper, entries of sequences, infinite series and infinite matrices are real or complex numbers. The class \((\ell_\alpha, \ell_\alpha)\), \(0 < \alpha \leq 1\), of all infinite matrices transforming sequences in \(\ell_\alpha\) to sequences in \(\ell_\alpha\) is characterized. The structure of \((\ell_\alpha, \ell_\alpha)\), \(0 < \alpha \leq 1\), is then discussed. Following Fridy [\emph{Properties of absolute summability matrices}, Proc. Amer. Math. Soc. 24 (1970), 583--585], a Steinhaus type result involving the class \((\ell_\alpha, \ell_\alpha)\) is also proved.

In this paper, K denotes a complete, non-trivially valued, non-archimedean (or ultrametric) field. Entries of double sequences, double series and 4-dimensional infinite matrices are in K.We prove Tauberian theorems for the Weighted Mean and (M,λ m,n ) methods for double series.

Throughout the present paper, K denotes a complete, non-trivially valued, ultrametric (or nonarchimedean) field. Sequences, infinite series and infinite matrices have their entries in K. The sequence spaces m?, c?, c?0 were introduced in K earlier by the author in [8-10] and some studies were made. The purpose of the present paper is to characterize the matrix classes (c?0, c? 0), (c?0,m?), (c?0, c?) and (c?, c?0).

Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered.For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings.It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitar associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks.A few open questions concerning properties of presented examples and other types of faithful representations are formulated.

Throughout this paper, entries of 4-dimensional infinite matrices, double sequences and double series are real or complex numbers. In the present paper, we introduce a new definition of convergence of a double sequence and a double series and record a few results on convergent double sequences. We also prove Silverman-Toeplitz theorem for double sequences and series.

In this paper, K denotes a complete, non-trivially valued, non-archimedean field. Infinite matrices, sequences and series have entries in K. In the present paper, which is a continuation of [4], we prove another interesting result concerning weighted means.

K denotes a complete, non-trivially valued, non-archimedean field. Infinite matrices, sequences and series have entries in K. In this paper, we prove an interesting result, which gives an equivalent formulation of summability by weighted mean methods. Incidentally this result includes the non-archimedean analogue of a theorem proved by Moricz and Rhoades (see [2], Theorem MR, p.188).

Throughout the present paper, entries of sequences, infinite series and infinite matrices are real or complex numbers. In this paper, we characterize the matrix class (??,??), 0 &lt; ? ? ? ? 1.

Let K be a complete, non-trivially valued, ultrametric (or non-archimedean) field, and λ = {λn} − a sequence in K with the property 0 &lt; |λn| ↗ ∞, n ➝ ∞, i.e., the speed of convergence. In the present paper, the concepts of speed-Maddox space, paranormed zero-convergence, paranormed convergence and paranormed boundedness with speed λ (or shortly, paranormed λ-zero-convergence, paranormed λ-convergence and paranormed λ-boundedness) over K have been recalled. Let μ be another speed in K. Necessary and sufficient conditions are found for a matrix A over K to transform the set of all paranormally λ-bounded sequences into the set of all paranormally μ-bounded, all paranormally μ-convergent or all paranormally μ-zero-convergent sequences.

Throughout this paper, K denotes a complete, non-trivially valued, ultrametric (or nonarchimedean) field. Sequences, infinite series and infinite matrices have entries in K. In this paper, we record some interesting properties about the matrix classes (c0,c0) and (c0,c0;P).

In this paper entries of sequences, series and infinite matrices are real or complex numbers. We introduce the (M,λn) method of summability and study some of its nice properties.