Jordan Journal of Mathematics and Statistics

Modern Functional Analysis in the Theory of Sequence Spaces and Matrix Transformations

2025-05-17T13:35:14+03:00

Many concepts and theories in functional analysis have turned out to be powerful and widely used tools in operator theory, in particular in the theory of matrix transformations between sequences spaces in summability. We give an introduction to the basic theory of FK, BK, AK and AD spaces, the various types of dual spaces of sequence spaces, and apply the general results to the characterisations of classes of matrix operators between certain sequence spaces that arise in summability. We also study the Hausdorff measure of noncompactness and its applications to the characterisations of compact operators between sequence spaces.
This is a survey paper which also includes some results of the author’s joint research with V. Rakoˇcevi´c and I. Djolevi´c at the Department of Mathematics of the Faculty of Science and Mathematics at the University of Niˇs, Serbia. Although many of the results are probably known to specialists, the proofs are included for the convenience of those readers who may not be too familiar with the subject, and an appendix is added at the end containing the fundamental theorems in functional analysis in the versions they are applied.

Key words and phrases. Summability, sequence spaces, FK, BK, AD, AK spaces, dual spaces, matrix transformations, measures of noncompactness.

1991 Mathematics Subject Classification. Primary: 40H05; Secondary: 46A15

Strongly Singular Calder´on-Zygmund Operators and their Commutators

2025-05-17T13:41:02+03:00

In this paper, the authors obtain two kinds of endpoint estimates for strongly singular Calder´on-Zygmund operators. Moreover, the pointwise estimate for the sharp maximal function of commutators generated by strongly singular Calder´on-Zygmund operators and BMO functions is also established. As its applications, the boundedness of the commutators on Morrey type spaces will be obtained.

Keywords: Strongly singular Calder´on-Zygmund operator, BMO, LMO, Commutator, Morrey space.

MR(2000) Subject Classification. 42B20, 42B35

Matrix Transformations Between Sets of the Form W- and Operator Generators of Analytic Semigroups

2025-05-17T13:46:10+03:00

In this paper we establish a relation between the notion of operators of analytic semigroups and matrix transformations from a set of sequences into w_∞. We get extensions of some results given by Labbas and de Malafosse concerning applications of the sum of operators in the nondifferential case.

Key words and phrases. Strongly bounded sequences, BK space, generators of analytic semigroup, Banach algebra, bounded operator.

1991 Mathematics Subject Classification. Primary: 35A35, Secondary 40C05, 46A45

Best Simultaneous Approximation in Metric Spaces

2025-05-17T15:51:17+03:00

For a Banach space X and an increasing subadditive continuous function φ on [0,∞) with φ(0) = 0, let us denote by L^φ(I,X), the space of all X-valued φ-integrable functions f : I → X on a certain positive complete σ-finite measure space (equation). The aim of this paper is to prove that for a closed separable subspace G of X, L^φ(I,G) is simultaneously proximinal in L^φ(I,X) if and only if G is simultaneously proximinal in X. Other result on simultaneous approximation of l^φ(G) in l^φ(X) is presented.

Key words and phrases. Simultaneous, Approximation.

1991 Mathematics Subject Classification. Primary: 41A65; Secondary: 41A50

Algebraic Models in Applied Research

2025-05-17T16:02:18+03:00

Mathematical models can be used in several sciences as a tool to organize the research in such a way that the results could be transferred into other sciences. Here we go on with a method how to organise some results, which do not seem to have any relation with mathematics in a strictly algebraic way. The topic of algebra used is called H_v-structures, the largest class of hyperstructures. Our examples are taken from recent research in linguistics, more specifically on sociolinguistics. We believe that this game with mathematics will stimulate the interest of both researchers and students; moreover it is within the frame of Interdisciplinary Approach, one of the most recent teaching attitudes.

Key words: hyperstructures, H_v-structures.

AMS Subject Classification. 20N20

The Set of Values of Functionals in the Classes of Functions Having Integral Representation

2025-05-17T16:08:11+03:00

This paper presents a certain method to determine the set of values of the functional (equation) defined on the family of analytic functions in the unit disc possessing an integral representation with kernel q(z,t) in the interval [α ,β ]. It has been shown that the set of values of the linear functional J[ f( z )] at the point z₀ is the convex hull of the curve Γ, given by (equation). The Class of Functions with Limited Rotation was cosidered as a special case and some new results were obtained.