A Certain Class of Statistical Deferred Weighted A-summability Based on (p; q)-integers and Associated Approximation Theorems

Statistical summability has recently enhanced researchers’ substantial awareness since it is more broad than the traditional (ordinary) convergence. The basic concept of statistical weighted A- summability was introduced by Mohiuddine (2016). In this investigation, we introduce the (presumably new) concept of statistical deferred weighted A-summability and deferred weighted A- statistical convergence with respect to the difference sequence of order r involving (p; q)-integers and establish an inclusion relation between them. Furthermore, based upon the proposed methods, we intend to approximate the rate of convergence and to demonstrate a Korovkin type approximation theorem for functions of two variables defined on a Banach space CB(D). Finally, several illustrative examples are presented in light of our definitions and outcomes established in this paper

Matemaatika- ja mehaanikaalaseid töid. Functional analysis and theory of summability

• Table of contents • Aasma. Characterization of matrix transformations of summability fields • Resümee • J. Arhippainen. On commutative locally m-convex algebras • Resümee • J. Boos, T. Leiger. Product and direct sum of L-K(X)-spaces and related K(X)-spaces • Resümee • E. Kolk. The statistical convergence in Banach space • Resümee • Lepasson. T-dual spaces with rate and T-sectionally summable spaces with rate in the case of double sequences • Resümee • L. Loone. On cores of semicontinuous sequential summability methods • Resümee • L. Loone. Inclusion between the cores concerning weighted means and power series • Resümee • Monakov-Rogozkin. A description of measure spaces with liftings • Resümee • E. Oja. Remarks on the dual of the space of continuous linear operators • Resümee • V. Soomer. Summability factors for strong summability • Resümee • H. Türnpu. Weyl factors for summability with speed of orthogonal series • Resümeehttp://tartu.ester.ee/record=b1078154~S1*es

Ideal-quasi-Cauchy sequences

An ideal $I$ is a family of subsets of positive integers $\textbf{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_n)$ of real numbers is said to be $I$-convergent to a real number $L$, if for each \;$\varepsilon> 0$ the set $\{n:|x_{n}-L|\geq \varepsilon\}$ belongs to $I$. We introduce $I$-ward compactness of a subset of $\textbf{R}$, the set of real numbers, and $I$-ward continuity of a real function in the senses that a subset $E$ of $\textbf{R}$ is $I$-ward compact if any sequence $(x_{n})$ of points in $E$ has an $I$-quasi-Cauchy subsequence, and a real function is $I$-ward continuous if it preserves $I$-quasi-Cauchy sequences where a sequence $(x_{n})$ is called to be $I$-quasi-Cauchy when $(\Delta x_{n})$ is $I$-convergent to 0. We obtain results related to $I$-ward continuity, $I$-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, $\delta$-ward continuity, and slowly oscillating continuity.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1005.494

Rainwater-Simons-type convergence theorems for generalized convergence methods

We extend the well-known Rainwater-Simons convergence theorem to various generalized convergence methods such as strong matrix summability, statistical convergence and almost convergence. In fact we prove these theorems not only for boundaries but for the more general notion of (I)-generating sets introduced by Fonf and Lindenstrauss.Comment: 10 pages, version 2, references added, one remark added, revised version accepted for publication in Acta et Commentationes Universitatis Tartuensis de Mathematic