Recent zbMATH articles in MSC 40https://zbmath.org/atom/cc/402022-11-17T18:59:28.764376ZWerkzeugRelation between matrices and the suborbital graphs by the special number sequenceshttps://zbmath.org/1496.110192022-11-17T18:59:28.764376Z"Akbaba, Ümmügülsün"https://zbmath.org/authors/?q=ai:akbaba.ummugulsun"Değer, Ali Hikmet"https://zbmath.org/authors/?q=ai:deger.ali-hikmetSummary: Continued fractions and their matrix connections have been used in many studies to generate new identities. On the other hand, many examinations have been made in the suborbital graphs under circuit and forest conditions. Special number sequences and special vertex values of minimal length paths in suborbital graphs have been associated in our previous studies. In these associations, matrix connections of the special continued fractions \(\mathcal{K}(-1/-k)\), where \(k\in\mathbb{Z}^+\), \(k\geq 2\) with the values of the special number sequences are used and new identities are obtained. In this study, by producing new matrices, new identities related to Fibonacci, Lucas, Pell, and Pell-Lucas number sequences are found by using both recurrence relations and matrix connections of the continued fractions. In addition, the farthest vertex values of the minimal length path in the suborbital graph \(\mathbf{F}_{u, N}\) and these number sequences are associated.Some properties of the remainders in certain series representations for the constant \(\pi\)https://zbmath.org/1496.111412022-11-17T18:59:28.764376Z"Han, Xue-Feng"https://zbmath.org/authors/?q=ai:han.xuefeng"Chen, Chao-Ping"https://zbmath.org/authors/?q=ai:chen.chaopingSummary: The constant \(\pi\) has many series representations. In this paper, we consider some properties of the remainders in certain series representations for the constant \(\pi\), including analytical representations, asymptotic expansions and inequalities. In 1963, \textit{J. S. Frame} [Problem 5113, Am. Math. Mon. 70, No. 6, 672 (1963)] proposed the following problem: Sum the series
\[
S=\sum_{k=0}^\infty \binom{2k}{k}(-16)^{-k}(2k+1)^{-2}.
\]
Subsequently, \textit{A. Weinmann} [Sum of an infinite series, Am. Math. Mon. 71, No. 6, 691--692 (1964] proved the required sum \(S=\frac{\pi^2}{10}\). We here establish the following integral representation of \(S\):
\[
S = 2\int_0^1\frac{1}{x}\ln\left(\frac{x}{2} + \sqrt{\frac{x^2}{4}+1}\right) \mathrm{d}x = -2\int_0^1\frac{\ln x}{\sqrt{x^2+4}}\mathrm{d}x = \frac{\pi^2}{10}.
\]Regular functions of a quaternionic variablehttps://zbmath.org/1496.300012022-11-17T18:59:28.764376Z"Gentili, Graziano"https://zbmath.org/authors/?q=ai:gentili.graziano"Stoppato, Caterina"https://zbmath.org/authors/?q=ai:stoppato.caterina"Struppa, Daniele C."https://zbmath.org/authors/?q=ai:struppa.daniele-carloPublisher's description: This book surveys the foundations of the theory of slice regular functions over the quaternions, introduced in 2006, and gives an overview of its generalizations and applications.
As in the case of other interesting quaternionic function theories, the original motivations were the richness of the theory of holomorphic functions of one complex variable and the fact that quaternions form the only associative real division algebra with a finite dimension \(n>2\). (Slice) regular functions quickly showed particularly appealing features and developed into a full-fledged theory, while finding applications to outstanding problems from other areas of mathematics. For instance, this class of functions includes polynomials and power series. The nature of the zero sets of regular functions is particularly interesting and strictly linked to an articulate algebraic structure, which allows several types of series expansion and the study of singularities. Integral representation formulas enrich the theory and are fundamental to the construction of a noncommutative functional calculus. Regular functions have a particularly nice differential topology and are useful tools for the construction and classification of quaternionic orthogonal complex structures, where they compensate for the scarcity of conformal maps in dimension four.
This second, expanded edition additionally covers a new branch of the theory: the study of regular functions whose domains are not axially symmetric. The volume is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general.
See the review of the first edition in [Zbl 1269.30001].Integral formula for the Bessel function of the first kindhttps://zbmath.org/1496.330042022-11-17T18:59:28.764376Z"De Micheli, Enrico"https://zbmath.org/authors/?q=ai:de-micheli.enricoSummary: In this paper, we prove a new integral formula for the Bessel function of the first kind \(J_\mu (z)\). This formula generalizes to any \(\mu ,z\in{\mathbb{C}}\) the classical integral representations of Bessel and Poisson.On asymptotically ideal invariant equivalence of double sequenceshttps://zbmath.org/1496.340812022-11-17T18:59:28.764376Z"Dündar, Erdinç"https://zbmath.org/authors/?q=ai:dundar.erdinc"Ulusu, Uğur"https://zbmath.org/authors/?q=ai:ulusu.ugur"Nuray, Fatih"https://zbmath.org/authors/?q=ai:nuray.fatihSummary: In this study, the concepts of asymptotically \(\mathcal{I}^{\sigma}_2\)-equivalent, asymptotically invariant equivalent, strongly asymptotically invariant equivalent and \(p\)-strongly asymptotically invariant equivalent for double sequences are defined. Also, we investigate relationships among these new type equivalence concepts.Statistical convergence in paranorm sense on time scaleshttps://zbmath.org/1496.341302022-11-17T18:59:28.764376Z"Gulsen, Tuba"https://zbmath.org/authors/?q=ai:gulsen.tuba"Koyunbakan, Hikmet"https://zbmath.org/authors/?q=ai:koyunbakan.hikmet"Yılmaz, Emrah"https://zbmath.org/authors/?q=ai:yilmaz.emrah-sercan"Altın, Yavuz"https://zbmath.org/authors/?q=ai:altin.yavuzSummary: In this study, we define statistical convergence and \(\lambda\)-statistical convergence in paranorm sense on anarbitrary time scale equipped with paranorm. Furthermore, we study on strongly \(\lambda_p\)-summability on timescales in paranorm sense. Eventually, some inclusion theorems are proved.On rough convergence in 2-normed spaces and some propertieshttps://zbmath.org/1496.400012022-11-17T18:59:28.764376Z"Arslan, Mukaddes"https://zbmath.org/authors/?q=ai:arslan.mukaddes"Dündar, Erdinç"https://zbmath.org/authors/?q=ai:dundar.erdincSummary: In this study, we investigated relationships between rough convergence and classical convergence and studied some properties about the notion of rough convergence, the set of rough limit points and rough cluster points of a sequence in 2-normed space. Also, we examined the dependence of \(r\)-limit LIM\(^r_2x_n\) of a fixed sequence \((x_n)\) on varying parameter \(r\) in 2-normed space.Summability of subsequences of a divergent sequence by regular matrices. II.https://zbmath.org/1496.400022022-11-17T18:59:28.764376Z"Boos, Johann"https://zbmath.org/authors/?q=ai:boos.johannSummary: \textit{C. Stuart} proved in [Rocky Mt. J. Math. 44, No. 1, 289--295 (2014; Zbl 1298.40007), Proposition 7] that the Cesàro matrix C 1 cannot sum almost every subsequence of a bounded divergent sequence. At the end of the paper he remarked `It seems likely that this proposition could be generalized for any regular matrix, but we do not have a proof of this'. In [\textit{J. Boos} and \textit{M. Zeltser}, Rocky Mt. J. Math. 48, No. 2, 413--423 (2018; Zbl 1400.40001), Theorem 3.1] Stuart's conjecture is confirmed, and it is even extended to the more general case of divergent sequences. In this note, we show that [loc. cit., Theorem 3.1] is a special case of Theorem 3.5.5 in [\textit{G. M. Petersen}, Regular matrix transformations. New York etc.: McGraw-Hill Publishing Company Ltd. (1966; Zbl 0159.35401)] by proving that the set of all index sequences with positive density is of the second category. For the proof of that a decisive hint was given to the author by Harry I. Miller a few months before he passed away on 17 December 2018.
For Part I see [\textit{J. Boos} and \textit{M. Zeltser}, Rocky Mt. J. Math. 48, No. 2, 413--423 (2018; Zbl 1400.40001)].Abel statistical quasi Cauchy sequenceshttps://zbmath.org/1496.400032022-11-17T18:59:28.764376Z"Cakalli, Huseyin"https://zbmath.org/authors/?q=ai:cakalli.huseyinSummary: In this paper, we investigate the concept of Abel statistical quasi Cauchy sequences. A real function \(f\) is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchy sequences, where a sequence \((\alpha_k)\) of point in \(\mathbb{R}\) is called Abel statistically quasi Cauchy if \(\lim_{x\to 1}-(1-x)\Sigma_{k:|\Delta\alpha_k|\geqslant\varepsilon}x^k=0\) for
every \(\varepsilon>0\), where \(\Delta\alpha_k=\alpha_{k+1}-\alpha_k\) for every \(k\in\mathbb{N}\). Some other types of continuities are also studied and interesting results are obtained. It turns out that the set of Abel statistical ward continuous functions is a closed subset of the space of continuous functions.A generalization of Orlicz sequence spaces derived by quadruple sequential band matrixhttps://zbmath.org/1496.400042022-11-17T18:59:28.764376Z"Dutta, Salila"https://zbmath.org/authors/?q=ai:dutta.salila"Tripathy, Nilambar"https://zbmath.org/authors/?q=ai:tripathy.nilambarSummary: In this article we have introduced a new Orlicz sequence space \(l_p^\lambda(M,B)\) derived by a quadruple sequential band matrix associated with an Orlicz function and lambda matrix. Further, we have studied some topological properties and inclusion relations of this space.Convergence follows from Cesàro summability in the case of slowly decreasing or slowly oscillating double sequences in certain senseshttps://zbmath.org/1496.400052022-11-17T18:59:28.764376Z"Önder, Zerrin"https://zbmath.org/authors/?q=ai:onder.zerrin"Çanak, İbrahim"https://zbmath.org/authors/?q=ai:canak.ibrahimSummary: Let \((u_{\mu\nu})\) be a double sequence of real or complex numbers which is \((C,1,1)\) summable to a finite limit. We obtain some Tauberian conditions of slow decreasing or oscillating types in terms of the generator sequences in certain senses under which \(P\)-convergence of a double sequence \((u_{\mu\nu})\) follows from its \((C,1,1)\) summability. We give Tauberian theorems in which Tauberian conditions are of Hardy and Landau types as special cases of our results. We present some Tauberian conditions in terms of the de la Vallée Poussin means of double sequences under which \(P\)-convergence of a double sequence \((u_{\mu\nu})\) follows from its \((C,1,1)\) summability. Moreover, we give analogous results for \((C,1,0)\) and \((C,0,1)\) summability methods.On rearrangements of infinite series with complex termshttps://zbmath.org/1496.400062022-11-17T18:59:28.764376Z"Simonič, Aleksander"https://zbmath.org/authors/?q=ai:simonic.aleksander|simonic.aleksander.1Summary: The Lévy-Steinitz theorem for conditionally convergent series with complex terms says that the set of sums we obtain after all rearrangements is either a line or the complex plane. In the article we present a detailed proof of this theorem.Infinite sums related to the generalized Fibonacci numbershttps://zbmath.org/1496.400072022-11-17T18:59:28.764376Z"Uslu, Kemal"https://zbmath.org/authors/?q=ai:uslu.kemal"Teke, Mustafa"https://zbmath.org/authors/?q=ai:teke.mustafaSummary: Fibonacci numbers and applications related to these numbers are frequently encountered both in daily life and in various fields of science and engineering. There are many studies to sum expressions on these numbers [\textit{T. Koshy}, Fibonacci and Lucas numbers with applications. Volume I. New York, NY: Wiley (2001; Zbl 0984.11010)]. However, in later periods, generalized Fibonacci numbers, which are the more general version of Fibonacci and Lucas numbers, and also new number sequences such as \(k\)-Fibonacci numbers by Sergio Falcon have entered into the literature [\textit{S. Falcón} and \textit{Á. Plaza}, Chaos Solitons Fractals 32, No. 5, 1615--1624 (2007; Zbl 1158.11306)]. In this study, some sums of generalized Fibonacci numbers have been investigated and compared with previously obtained sums of Fibonacci and Lucas numbers, which are the special cases of these sums.On statistical and strong convergence with respect to a modulus function and a power series methodhttps://zbmath.org/1496.400082022-11-17T18:59:28.764376Z"Belen, Cemal"https://zbmath.org/authors/?q=ai:belen.cemal"Yıldırım, Mustafa"https://zbmath.org/authors/?q=ai:yildirim.mustafa"Sümbül, Canan"https://zbmath.org/authors/?q=ai:sumbul.cananSummary: This paper introduces and focuses on two pairs of concepts in two main sections. The first section aims to examine the relation between the concepts of strong \(J_p\)-convergence with respect to a modulus function \(f\) and \(J_p\)-statistical convergence, where \(J_p\) is a power series method. The second section introduces the notions of \(f\)-\(J_p\)-statistical convergence and \(f\)-strong \(J_p\)-convergence and discusses some possible relations among them.Influence of \(\theta\)-metric spaces on the diameter of rough weighted \(\mathcal{I}_2\)-limit sethttps://zbmath.org/1496.400092022-11-17T18:59:28.764376Z"Ghosal, Sanjoy"https://zbmath.org/authors/?q=ai:ghosal.sanjoy-kr"Listán-García, M. C."https://zbmath.org/authors/?q=ai:listan-garcia.m-c"Mandal, Manasi"https://zbmath.org/authors/?q=ai:mandal.manasi"Banerjee, Mandobi"https://zbmath.org/authors/?q=ai:banerjee.mandobiSummary: In this paper we continue our investigation of the recent summability notion introduced in [\textit{S. Ghosal} and \textit{A. Ghosh}, Math. Slovaca 69, No. 4, 871--890 (2019; Zbl 07289565)] (where rough weighted statistical convergence for double sequences is discussed over norm linear spaces) and introduce the notion of rough weighted \(\mathcal{I}_2\)-convergence over \(\theta\)-metric spaces. Also we exercise the behavior of weighted \(\mathcal{I}_2\)-cluster points set over \(\theta\)-metric spaces. Based on the new notion, we discuss some important results and perceive how the existing results are vacillating.Conglomerated filters and statistical measureshttps://zbmath.org/1496.400102022-11-17T18:59:28.764376Z"Kadets, Vladimir"https://zbmath.org/authors/?q=ai:kadets.vladimir-m"Seliutin, Dmytro"https://zbmath.org/authors/?q=ai:seliutin.dmytro"Tryba, Jacek"https://zbmath.org/authors/?q=ai:tryba.jacekThe paper under review is an interdisciplinary paper which deals with measure-theoretical problems connected to statistical convergence, from the point of view of analysis, and with filters and ultrafilters, from a set-theoretical point of view.
Ultrafilters have been introduced by \textit{F.~Riesz} [``Stetigkeitsbegriff und abstrakte Mengenlehre'', Atti del IV Congresso Intern. Matem., Roma 1908, II, Roma, 18--24 (1909; JFM 40.0098.07)], see [\textit{H. L. Bentley} et al., in: Handbook of the history of general topology. Volume 2. Dordrecht: Kluwer Academic Publishers. 577--629 (1998; Zbl 0936.54028)], but have not received due attention until they were rediscovered by the French school.
In the terminology of the paper under review, a \emph{statistical measure} is a non-negative finitely additive measure \(\mu \) defined on the collection of all subsets of $\mathbb N$ and such that \( \mu (\mathbb N) = 1\) and \( \mu (\{ k \} ) = 0\), for all \(k \in \mathbb N\). The name is motivated by the notion of statistical convergence, which plays an important role in mathematical analysis, measure theory and functional analysis. Notice that a \(\{ 0,1\}\)-valued statistical measure corresponds to a free ultrafilter over \(\mathbb N\). As pointed out by the authors, statistical measures, without using this name, have been considered earlier by other scholars working in axiomatic set theory, model theory and descriptive set theory.
The \emph{filter generated by a statistical measure} \(\mu \) is the collection of all subsets \(A \subseteq \mathbb N\) such that \(\mu (A)=1\). With the aim of characterizing such filters, the authors introduce the notions of a \emph{poor} and of a \emph{conglomerated} filter. They show that every filter generated by a statistical measure is poor, that every conglomerated filter is not poor, and that there is a filter which is neither poor nor conglomerated. A filter with the Baire property is conglomerated, hence it is not generated by a statistical measure.
Many examples of statistical measures are known such that the corresponding filter cannot be represented as a countable intersection of ultrafilters. In Section~3, the authors study intersections of families of ultrafilters. If some filter can be represented as a finite intersection of ultrafilters, the representation is unique; this is not the case for infinite intersections.
In Section 4, among other things, an extract from a letter by Piotr Koszmider is reproduced, showing that there is a poor filter which is not generated by a statistical measure. The argument relies on Boolean algebra techniques. The authors credit valuable comments also to István Juhász.
This well-written paper will surely foster further the collaboration among analysts and set theorists.
Reviewer: Paolo Lipparini (Roma)Lacunary \(d\)-statistical boundedness of order \(\alpha\) in metric spaceshttps://zbmath.org/1496.400112022-11-17T18:59:28.764376Z"Kandemir, H. Şengül"https://zbmath.org/authors/?q=ai:kandemir.hacer-sengul"Et, M."https://zbmath.org/authors/?q=ai:et.mikail"Çakallı, H."https://zbmath.org/authors/?q=ai:cakalli.huseyinSummary: In this study, using a lacunary sequence we introduce the concepts of lacunary \(d\)-statistically convergent sequences of order \(\alpha\) and lacunary \(d\)-statistically bounded sequences of order \(\alpha\) in general metric spaces.On ideal convergence of triple sequences in intuitionistic fuzzy normed space defined by compact operatorhttps://zbmath.org/1496.400122022-11-17T18:59:28.764376Z"Khan, Vakeel A."https://zbmath.org/authors/?q=ai:khan.vakeel-a"Idrisi, Mohd. Imran"https://zbmath.org/authors/?q=ai:idrisi.mohd-imran"Tuba, Umme"https://zbmath.org/authors/?q=ai:tuba.ummeSummary: The main purpose of this article is to introduce and study some new spaces of \(I\)-convergence of triple sequences in intuitionistic fuzzy normed space defined by compact operator i.e., \(_3 S^I_{(\mu,\nu)}(T)\) and \(_3 S^I_{0(\mu,\nu)}(T)\) and examine some fundamental properties, fuzzy topology and verify inclusion relations lying under these spaces.A new generalized version of Korovkin-type approximation theoremhttps://zbmath.org/1496.400132022-11-17T18:59:28.764376Z"Khan, Vakeel A."https://zbmath.org/authors/?q=ai:khan.vakeel-a"Khan, Izhar Ali"https://zbmath.org/authors/?q=ai:khan.izhar-ali"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipanIn statistical convergence, the convergence condition is obtained just for a majority of elements. Therefore it extends the concept of ordinary convergence and it is an effective tool to obtain strong results. Recently, there have been obtained many generalizations of statistical convergence by combining it with ideal, measure and mean.
In this paper, the authors introduce \(\mu\)-statistical measurable convergence, ideal \(\mu\)-statistical measurable convergence, \(\mu\)-deferred ideal statistical measurable convergence, \(\mu\)-deferred ideal statistical mean convergence of order \(\alpha\), \(\mu\)-deferred ideal statistical mean convergence in measure of order \(\alpha\), \(\mu\)-deferred statistical Lebesgue measurable convergence under integral. They examine the implications between these concepts of convergences in detail. They also provide examples for the converse parts. At the end, as an application they present Korovkin-type results with the use of these notions.
Reviewer: Tuğba Yurdakadim (Bilecik)A new type of paranorm intuitionistic fuzzy Zweier \(I\)-convergent double sequence spaceshttps://zbmath.org/1496.400142022-11-17T18:59:28.764376Z"Khan, Vakeel A."https://zbmath.org/authors/?q=ai:khan.vakeel-a"Yasmeen"https://zbmath.org/authors/?q=ai:yasmeen.saba|yasmeen.k-zeba|yasmeen.uzma|yasmeen.farah|yasmeen.shagufta|yasmeen.adeela|yasmeen.hafsa"Fatima, Hira"https://zbmath.org/authors/?q=ai:fatima.hira"Altaf, Henna"https://zbmath.org/authors/?q=ai:altaf.hennaSummary: In this article we introduce the paranorm type intuitionistic fuzzy Zweier \(I\)-convergent double sequence spaces \(_2\mathcal{Z}^I_{(\mu,\nu)}(p)\) and \(_2\mathcal{Z}^I_{0(\mu,\nu)}(p)\) for \(p=(p_{ij})\) a double sequence of positive real numbers and study the fuzzy topology on these spaces.\((A, \varphi)\)-lacunary statistical convergence of order \(\alpha\)https://zbmath.org/1496.400152022-11-17T18:59:28.764376Z"Savaş, Ekrem"https://zbmath.org/authors/?q=ai:savas.ekremSummary: In the present paper, we introduce and study \((A,\varphi)\)-statistical convergence of order \(\alpha\), using the \(\varphi\)-function, infinite matrix and we establish some inclusion theorems.On \(I\)-deferred statistical convergence of order \(\alpha\)https://zbmath.org/1496.400162022-11-17T18:59:28.764376Z"Şengül, Hacer"https://zbmath.org/authors/?q=ai:sengul.hacer"Et, Mikail"https://zbmath.org/authors/?q=ai:et.mikail"Işık, Mahmut"https://zbmath.org/authors/?q=ai:isik.mahmutSummary: The idea of \(I\)-convergence of real sequences was introduced by \textit{P. Kostyrko} et al. [Real Anal. Exch. 26, No. 2, 669--685 (2001; Zbl 1021.40001)] and also independently by \textit{F. Nuray} and \textit{W. H. Ruckle} [J. Math. Anal. Appl. 245, No. 2, 513--527 (2000; Zbl 0955.40001)]. In this paper we introduce \(I\)-deferred statistical convergence of order \(\alpha\) and strong \(I\)-deferred Cesàro convergence of order \(\alpha\) and investigated between their relationship.Solvability of some perturbed sequence spaces equations with operatorshttps://zbmath.org/1496.400172022-11-17T18:59:28.764376Z"de Malafosse, Bruno"https://zbmath.org/authors/?q=ai:de-malafosse.bruno"Fares, Ali"https://zbmath.org/authors/?q=ai:fares.ali"Ayad, Ali"https://zbmath.org/authors/?q=ai:ayad.aliSummary: In this paper, we apply the results stated in [\textit{B. de Malafosse} et al., Filomat 32, No. 14, 5123--130 (2018; Zbl 07552731)] to the solvability of the sequence spaces equations (SSE) \(\mathcal{E}+F_x=F_b\), where \(\mathcal{E},F\) are linear spaces of sequences and \(b,x\) are positive sequences (\(x\) is the unknown). In this way, we solve the (SSE) of the form \((E_a)_{G(\alpha,\beta)}+F_x=F_b\), where \(G(\alpha,\beta)\) is a factorable triangle matrix defined by \([G(\alpha, \beta)]_{nk}=\alpha_n\beta_k\) for \(k\leq n\) and \((E,F)\in\{(\ell_\infty,c),(c_0,\ell_\infty),(c_0,c),(\ell^p,c),(\ell^p,\ell_\infty),(w_0,\ell_\infty)\}\) with \(p\geq 1\). Then we deal with some (SSE) involving the matrices \(C(\lambda)\), \(C_1\) and \(\overline{N}_q\). Finally, we solve the (SSE) with operator of the form \((E_a)_{\Sigma^2}+F_x=F_b\).Tauberian conditions under which convergence follows from Cesàro summability of double integrals over \(\mathbb{R}_+^2\)https://zbmath.org/1496.400182022-11-17T18:59:28.764376Z"Fındık, Gökşen"https://zbmath.org/authors/?q=ai:findik.goksen"Çanak, İbrahim"https://zbmath.org/authors/?q=ai:canak.ibrahimSummary: For a real- or complex-valued continuous function \(f\) over \(\mathbb{R}^2_+:=[0,\infty)\times[0,\infty)\), we denote its integral over \([0,u]\times [0,v]\) by \(s(u,v)\) and its \((C,1,1)\) mean, the average of \(s(u,v)\) over \([0,u]\times [0,v]\), by \(\sigma(u,v)\). The other means \((C,1,0)\) and \((C,0,1)\) are defined analogously. We introduce the concepts of backward differences and the Kronecker identities in different senses for double integrals over \(\mathbb{R}^2_+\). We give onesided and two-sided Tauberian conditions based on the difference between double integral of \(s(u,v)\) and its means in different senses for Cesàro summability methods of double integrals over \([0,u]\times[0,v]\) under which convergence of \(s(u,v)\) follows from integrability of \(s(u,v)\) in different senses.On E-J Hausdorff transformations for double sequenceshttps://zbmath.org/1496.400192022-11-17T18:59:28.764376Z"Savaş, Rabia"https://zbmath.org/authors/?q=ai:savas.rabia"Şevli, Hamdullah"https://zbmath.org/authors/?q=ai:sevli.hamdullahSummary: \textit{C. R. Adams} [Bull. Am. Math. Soc. 39, 303--312 (1933; Zbl 0007.11701)] developed Hausdorff transformations for double sequences. \textit{H. Şevli} and \textit{R. Savaş} [J. Inequal. Appl. 2014, Paper No. 240, 10 p. (2014; Zbl 1327.40004)] proved some result for the double Endl-Jakimovski (E-J) generalization. In this study, we consider some further results for E-J Hausdorff transformations for double sequences.\(G\)-connectedness in topological groups with operationshttps://zbmath.org/1496.400202022-11-17T18:59:28.764376Z"Mucuk, Osman"https://zbmath.org/authors/?q=ai:mucuk.osman"Çakallı, Hüseyin"https://zbmath.org/authors/?q=ai:cakalli.huseyinSummary: It is a well-known fact that for a Hausdorff topological group \(X\), the limits of convergent sequences in \(X\) define a function denoted by \(\lim\) from the set of all convergent sequences in \(X\) to \(X\). This notion has been modified by \textit{J. Connor} and \textit{K. G. Grosse-Erdmann} [Rocky Mt. J. Math. 33, No. 1, 93--121 (2003; Zbl 1040.26001)] for real functions by replacing \(\lim\) with an arbitrary linear functional \(G\) defined on a linear subspace of the vector space of all real sequences. Recently, some authors have extended the concept to the topological group setting and introduced the concepts of \(G\)-continuity, \(G\)-compactness and \(G\)-connectedness. In this paper, we present some results about \(G\)-hulls, \(G\)-connectedness and \(G\)-fundamental systems of \(G\)-open neighbourhoods for a wide class of topological algebraic structures called groups with operations, which include topological groups, topological rings without identity, R-modules, Lie algebras, Jordan algebras, and many others.Some properties of Kantorovich variant of Chlodowsky-Szász operators induced by Boas-Buck type polynomialshttps://zbmath.org/1496.410092022-11-17T18:59:28.764376Z"Braha, Naim L."https://zbmath.org/authors/?q=ai:braha.naim-latif"Loku, Valdete"https://zbmath.org/authors/?q=ai:loku.valdete"Mansour, Toufik"https://zbmath.org/authors/?q=ai:mansour.toufikIn this paper the authors have introduced the Kantorovich variant of Chlodowsky-Szász operators inspred by Boas-Buck type polynomials which is introduced by \textit{I. Chlodovsky} [Compos. Math. 4, 380--393 (1937; Zbl 0016.35401)], and have examined some properties of the new operators. By using modulus of continuity and Peetre's K-functional, some results about rate of convergence have been given for the operators. Also Voronovskaya type result the operators has been stated. On the other hand weighted versions of the above results have been examined.
Reviewer: Emre Taş (Kırşehir)Triangular Cesàro summability and Lebesgue points of two-dimensional Fourier serieshttps://zbmath.org/1496.420112022-11-17T18:59:28.764376Z"Weisz, Ferenc"https://zbmath.org/authors/?q=ai:weisz.ferencThe author proved that the triangular Cesàro means (of positive order) of bivariable functions \(f\in L^{p}(T^{2})\) with \(1\leq p<\infty \) converge to \(f\) at each strong \((1,\omega )\)-Lebesgue point. This generalizes the well-known classical Lebesgue's theorem.
Reviewer: Włodzimierz Łenski (Poznań)