Recent zbMATH articles in MSC 43https://zbmath.org/atom/cc/432021-01-08T12:24:00+00:00Werkzeug\(\mathcal{I}_H\)-regular Borel measures on locally compact abelian groups.https://zbmath.org/1449.430032021-01-08T12:24:00+00:00"Klotz, L."https://zbmath.org/authors/?q=ai:klotz.lukasz|klotz.lutz-peter|klotz.lawrence-h"Medina, J. M."https://zbmath.org/authors/?q=ai:medina.jose-m-moral|medina.juan-miguelLet \(G\) be an LCA group, \(H\) a closed subgroup, \(\Gamma\) the dual group of \(G\) and \(\Lambda\) the annihilator group of \(H\) in \(\Gamma\). Let \(\pi_H\) be the canonical homeomorphism from \(G\) onto the factor group \(G/H\) and \(\tilde{x} := \pi_H(x)\). For a non-empty subset \(S\) in \(G\) let \(P(S)\) denote the linear space of all trigonometric \(S\)-polynomials and let \(P(x+H) =: P(\tilde{x})\), \(x \in G\). Moreover let \(\mu\) be a regular finite non-negative measure on the Borel \(\sigma\)-algebra \(B(\Gamma)\). The measure \(\mu\) is called \(J_H\)-regular if and only if \[ \bigcap_{x\in G} C_\alpha P(x + H) = \bigcap_{\tilde{x}\in G/H}C_\alpha P(\tilde{x}) = \{0\}\] and is called \(J_H\)-singular if \(C_\alpha P(\tilde{x}) = L^\alpha(\mu)\). Here \(C_\alpha P(\tilde{x})\) denotes the closure of \(P(\tilde{x})\) in \(L^\alpha(\mu)\).
A characterization of \(J_H\)-regular measures is given in terms of be Radon-Nikodym derivatives of some measures defined by elements of the annihilator. Moreover the Wold type decomposition is obtained and relations to the Whittaker-Shannon-Kotel'nikov theorem are discussed.
Reviewer: Leszek Skrzypczak (Poznań)Johnson pseudo-contractibility of certain Banach algebras and their nilpotent ideals.https://zbmath.org/1449.430042021-01-08T12:24:00+00:00"Askari-Sayah, M."https://zbmath.org/authors/?q=ai:askari-sayah.m"Pourabbas, A."https://zbmath.org/authors/?q=ai:pourabbas.abdolrasoul"Sahami, A."https://zbmath.org/authors/?q=ai:sahami.amirSummary: In this paper, we study the notion of Johnson pseudo-contractibility for certain Banach algebras. For a bicyclic semigroup \(S\), we show that \(\ell^1(S)\) is not Johnson pseudo-contractible. Also for a Johnson pseudo-contractible Banach algebra \(A\), we show that \(A\) has no non-zero complemented closed nilpotent ideal.Generalized dyadic derivative and uniform convergence of its Walsh-Fourier series.https://zbmath.org/1449.420492021-01-08T12:24:00+00:00"Golubov, Boris I."https://zbmath.org/authors/?q=ai:golubov.boris-i"Volosivets, Sergey S."https://zbmath.org/authors/?q=ai:volosivets.sergey-sergeevichSummary: In the paper the notion of dyadic \(\lambda\)-derivative is introduced for the nonnegative, nondecreasing and concave sequence \(\{\lambda_n\}_{n=0}^{\infty}\). Analogues of Bernstein inequality for Walsh polynomials and of inverse approximation theorem are established. Also the uniform convergence of Walsh-Fourier series to this \(\lambda\)-derivative is studied.Amenability and Fan-Glicksberg theorem for set-valued mappings.https://zbmath.org/1449.430012021-01-08T12:24:00+00:00"Lau, Anthony To-Ming"https://zbmath.org/authors/?q=ai:lau.anthony-to-ming"Yao, Liangjin"https://zbmath.org/authors/?q=ai:yao.liangjinSummary: In this paper, we begin by discussion of some well known results on the existence of left invariant means in the spaces: \(LUC(S), AP(S)\) and \(WAP(S)\) with Hahn-Banach extension theorem. We then give a new and precise proof of the well known Fan-Glicksberg fixed point theorem. This is then followed by a discussion on some related open problems.Wavelet tight frames in Walsh analysis.https://zbmath.org/1449.420552021-01-08T12:24:00+00:00"Farkov, Yuri A."https://zbmath.org/authors/?q=ai:farkov.yuri-aSummary: We describe two type of wavelet tight frames associated with the generalized Walsh functions: (1) Parseval frames for \(L^2\)-spaces on Vilenkin groups, (2) finite tight frames for the space \(\ell^2(\mathbb{Z}_N)\). In particular cases these tight frames coincide with orthogonal wavelet bases associated with the classical Walsh functions.Fractional calculus and integral transforms of the \(M\)-Wright function.https://zbmath.org/1449.330092021-01-08T12:24:00+00:00"Khan, N. U."https://zbmath.org/authors/?q=ai:khan.nabiullah-u"Kashmin, T."https://zbmath.org/authors/?q=ai:kashmin.t"Khan, S. W."https://zbmath.org/authors/?q=ai:khan.shorab-waliSummary: This paper is concerned to investigate \(M\)-Wright function, which was earlier known as transcendental function of the Wright type. \(M\)-Wright function is a special case of the Wright function given by British mathematician (E. Maitland Wright) in 1933. We have explored the cosequences of Riemann-Liouville Integral and Differential operators on \(M\)-Wright function. We have also evaluated integral transforms of the \(M\)-Wright function.The Hardy-Dirichlet space \(\mathcal{H}^p\) and its composition operators.https://zbmath.org/1449.320012021-01-08T12:24:00+00:00"Queffélec, Hervé"https://zbmath.org/authors/?q=ai:queffelec.herveSummary: We present some recent results on composition operators acting on a Hardy space \(\mathcal{H}^p\) of a new type, formed by Dirichlet series. This study was initiated by Hedenmalm-Lindqvist-Seip for \(p = 2\) to answer a question of Beurling, and then it was continued for \(p\neq 2\) by Bayart. We get new results on the spectrum and the approximation numbers of such operators, especially when \(p = 1\). The proofs use interpolation sequences, Carleson measures and extensions of the Weyl inequalities to the Banach space setting, as well as the prime number theorem. Many interesting problems remain open.
For the entire collection see [Zbl 1404.42002].Boundedness of Littlewood-Paley operators with variable kernel on the weighted Herz-Morrey spaces with variable exponent.https://zbmath.org/1449.430022021-01-08T12:24:00+00:00"Abdalmonem, Afif"https://zbmath.org/authors/?q=ai:abdalmonem.afif"Abdalrhman, Omer"https://zbmath.org/authors/?q=ai:abdalrhman.omer"Tao, Shuangping"https://zbmath.org/authors/?q=ai:tao.shuangpingSummary: Let \(\Omega\in L^\infty(\mathbb{R}^n)\times L^2(S^{n-1})\) be a homogeneous function of degree zero. In this article, we obtain some boundedness of the parameterized Littlewood-Paley operators with variable kernels on weighted Herz-Morrey spaces with variable exponent. As a supplement, the boundedness of fractional integral operators with variable kernel is also obtained on these spaces.Pairs of dual wavelet frames on local fields.https://zbmath.org/1449.420522021-01-08T12:24:00+00:00"Bhat, M. Younus"https://zbmath.org/authors/?q=ai:bhat.mohammad-younusThe author introduces the notion of orthogonal wavelet frames on local fields of positive characteristic and presents an algorithm for the construction of a pair of orthogonal wavelet frames based on polyphase matrices formed by the polyphase components of the wavelet masks. He also gives a general construction algorithm for all orthogonal wavelet tight frames on local fields of positive characteristic from a compactly supported scaling function and investigates their properties by means of the Fourier transform. The motivation for this work are the papers by \textit{F. A. Shah} [Acta Univ. Apulensis, Math. Inform. 49, 47--65 (2017; Zbl 1413.42060)] on orthogonal wavelet frames generated by Walsh polynomials, and \textit{F. A. Shah} and \textit{L. Debnath} [Analysis, München 33, No. 3, 293--307 (2013; Zbl 1277.42047)] on tight wavelet frames on local fields.
Reviewer: Richard A. Zalik (Auburn)Dynamics behavior for second-order neutral Clifford differential equations: inertial neural networks with mixed delays.https://zbmath.org/1449.342412021-01-08T12:24:00+00:00"Aouiti, Chaouki"https://zbmath.org/authors/?q=ai:aouiti.chaouki"Ben Gharbia, Imen"https://zbmath.org/authors/?q=ai:ben-gharbia.imenSummary: In this paper, Clifford-valued inertial neutral neural networks with time-varying delays and infinite distributed delay are investigated. With the help of the pseudo almost periodic function theory, Banach's fixed point theorem, and the differential inequality theory, a set of sufficient conditions that guarantee the existence and the global exponential stability of unique pseudo-almost periodic solutions of Clifford-valued inertial neutral neural networks with mixed delays are established. Our results are new and complement some previously known ones. Moreover, numerical simulations are carried out to illustrate our theoretical results.Quasi-asymptotically almost periodic vector-valued generalized functions.https://zbmath.org/1449.460302021-01-08T12:24:00+00:00"Kostić, Marko"https://zbmath.org/authors/?q=ai:kostic.marko"Pilipović, Stevan"https://zbmath.org/authors/?q=ai:pilipovic.stevan-r"Velinov, Daniel"https://zbmath.org/authors/?q=ai:velinov.danielSummary: In this paper are introduced the notions of quasi-asymptotically almost periodic distributions and quasi-asymptotically almost periodic ultradistributions with values in a Banach space, as well as some other generalizations of these concepts. Furthermore, some applications of the introduced concepts in the analysis of systems of ordinary differential equations are provided.The Fourier transform on 2-step Lie groups.https://zbmath.org/1449.430052021-01-08T12:24:00+00:00"Lévy, Guillaume"https://zbmath.org/authors/?q=ai:levy.guillaumeThe author develops harmonic analysis on a nilpotent Lie group of step 2. The Fourier transform is expressed in terms of the canonical bilinear form and its matrix coefficients. The parameter space of these matrix coefficients and its completion with respect to a natural distance are computed explicitly, as well as the integral kernel of the matrix coefficients Fourier transform, the analogue for the above framework of the classical Fourier kernel \((x, \xi)\mapsto e^{i(x\cdot \xi)}\).
Reviewer: Anatoly N. Kochubei (Kyïv)Unbounded translation invariant operators on commutative hypergroups.https://zbmath.org/1449.430062021-01-08T12:24:00+00:00"Kumar, Vishvesh"https://zbmath.org/authors/?q=ai:kumar.vishvesh"Kumar, N. Shiravan"https://zbmath.org/authors/?q=ai:kumar.n-shiravan"Sarma, Ritumoni"https://zbmath.org/authors/?q=ai:sarma.ritumoniLet \(K\) be a commutative hypergroup. The authors characterize translation invariant operators on \(L^1(K)\) and \(L^2(K)\) in terms of the Fourier transform. For these two cases, the space of all closed translation invariant operators forms a commutative algebra over \(\mathbb{C}\). An interpolation theorem for translation invariant operators on \(L^p(K)\), \(1\le p\le 2\), is proved.
Reviewer: Anatoly N. Kochubei (Kyïv)On a characterization theorem on non-discrete totally disconnected locally compact fields.https://zbmath.org/1449.600052021-01-08T12:24:00+00:00"Feldman, Gennadiy M."https://zbmath.org/authors/?q=ai:feldman.gennadiy-m"Myronyuk, Margaryta V."https://zbmath.org/authors/?q=ai:myronyuk.margarytaSummary: We prove the following theorem. Let \(X\) be a non-discrete totally disconnected locally compact field, \(R\) be its ring of integers, \(P\) be the nonzero prime ideal of \(R\). Assume that the residue field \(R/P\) is a field of characteristic \(p > 2\). Let \(\xi\) and \(\eta\) be independent identically distributed random variables with values in \(X\) and distribution \(\mu\), such that \(\mu\) has a continuous density with respect to a Haar measure on \(X\). This implies that the random variables \(S = \xi + \eta\) and \(D = (\xi-\eta)^2\) are independent if and only if \(\mu\) is a shift of the Haar distribution of a compact subgroup of \(X\).Almost automorphic solutions for shunting inhibitory cellular neural networks with leakage delays on time scales.https://zbmath.org/1449.343182021-01-08T12:24:00+00:00"Dai, Lihua"https://zbmath.org/authors/?q=ai:dai.lihua"Hui, Yuanxian"https://zbmath.org/authors/?q=ai:hui.yuanxianSummary: Shunting inhibitory cellular neural networks with time-varying delays in the leakage term and continuously distributed delays on a time scale \(T\) are proposed. Based on the exponential dichotomy of linear dynamic equation on time scales, fixed point theorems on time scales, we obtain some new sufficient conditions for the existence and global exponential stability of almost automorphic solution for the class of neural networks. Moreover, we give convictive numerical examples to show the feasibility of our results. This paper studies several classes of functional differential equations, including the existence of solutions and the stability of this solution on time scales.Pseudo almost automorphic solutions of hematopoiesis model with mixed delays.https://zbmath.org/1449.342842021-01-08T12:24:00+00:00"Aouiti, Chaouki"https://zbmath.org/authors/?q=ai:aouiti.chaouki"Dridi, Farah"https://zbmath.org/authors/?q=ai:dridi.farah"Kong, Fanchao"https://zbmath.org/authors/?q=ai:kong.fanchaoSummary: This paper is concerned with a hematopoiesis model with mixed delays. Under new conditions, we study the existence, uniqueness and global exponential stability of pseudo almost automorphic solutions for the suggested model. Our approach is mainly based on the exponential dichotomy of linear differential equation, Banach's fixed-point principle and suitable Lyapunov functional. At the end, some numerical examples are presented to demonstrate the effectiveness of our findings.Completions of quantum group algebras in certain norms and operators which commute with module actions.https://zbmath.org/1449.460602021-01-08T12:24:00+00:00"Nemati, Mehdi"https://zbmath.org/authors/?q=ai:nemati.mehdiSummary: Let \(L^1_{\text{cb}}(\mathbb{G})\) (respectively \(L^1_{\text{M}}(\mathbb{G})\)) denote the closure of the quantum group algebra \(L^1(\mathbb{G})\) of a locally compact quantum group \(\mathbb{G}\), in the space of completely bounded (respectively bounded) double centralizers of \(L^1(\mathbb{G}\)). In this paper, we study quantum group generalizations of various results from Fourier algebras of locally compact groups. In particular, left invariant means on \(L^1_{\text{cb}}(\mathbb{G})^*\) and on \(L^1_{\text{M}}(\mathbb{G})^*\) are defined and studied. We prove that the set of left invariant means on \(L^\infty(\mathbb{G})\) and on \(L^1_{\text{cb}}(\mathbb{G})^*(L^1_{\text{M}}(\mathbb{G})^*\)) have the same cardinality. We also study the left uniformly continuous functionals associated with these algebras. Finally, for a Banach \(\mathcal{A}\)-bimodule \(\mathcal{X}\) of a Banach algebra \(\mathcal{A}\) we establish a contractive and injective representation from the dual of a left introverted subspace of \(\mathcal{A}^*\) into \(B_\mathcal{A}(\mathcal{X}^*)\). As an application of this result we show that if the induced representation \(\varPhi:L\mathcal{U}C_{\text{cb}}(\mathbb{G})^*\to B_{L^1_{\text{cb}}(\mathbb{G})}(L^\infty(\mathbb{G}))\) is surjective, then \(L^1_{\text{cb}}(\mathbb{G})\) has a bounded approximate identity. We also obtain a characterization of co-amenable quantum groups in terms of representations of quantum measure algebras \(M(\mathbb{G})\).On some properties of relative capacity and thinness in weighted variable exponent Sobolev spaces.https://zbmath.org/1449.320122021-01-08T12:24:00+00:00"Unal, C."https://zbmath.org/authors/?q=ai:unal.cihan|unal.cemal"Aydin, I."https://zbmath.org/authors/?q=ai:aydin.ilknur|aydin.ismailLet \(p:\mathbb{R}^n\longrightarrow[1,+\infty)\) be a measurable function and let \(\vartheta:\mathbb{R}^n\longrightarrow(0,+\infty)\) be locally integrable. Denote by \(L^p_\vartheta(\mathbb{R}^n)\) the space of all measurable functions \(f\) such that \(\int_{\mathbb{R}^n}|f(x)|^{p(x)}\vartheta(x)dx<+\infty\) and let \(W^{1,p}_\vartheta(\mathbb{R}^n):=\{f\in L^p_\vartheta(\mathbb{R}^n): \partial{f}/\partial{x_j}\in L^p_\vartheta(\mathbb{R}^n),\;j=1,\dots,n\}\). The authors study the space \(W^{1,p}_\vartheta(\mathbb{R}^n)\) and various capacities associated with this space.
Reviewer: Marek Jarnicki (Kraków)