Recent zbMATH articles in MSC 40https://www.zbmath.org/atom/cc/402021-04-16T16:22:00+00:00WerkzeugOn Tauberian theorems for statistical weighted mean method of summability.https://www.zbmath.org/1456.400082021-04-16T16:22:00+00:00"Totur, Ümit"https://www.zbmath.org/authors/?q=ai:totur.umit"Çanak, İbrahim"https://www.zbmath.org/authors/?q=ai:canak.ibrahimSummary: In this paper we establish some new Tauberian theorems for the statistical weighted mean method of summability via the weighted general control modulo of the oscillatory behavior of nonnegative integer order of a real sequence.
The main results improve the well-known classical Tauberian theorems which are given for weighted mean method of summability and statistical convergence.Some Tauberian conditions under which convergence follows from \((C, 1, 1, 1)\) summability.https://www.zbmath.org/1456.400092021-04-16T16:22:00+00:00"Totur, Ümit"https://www.zbmath.org/authors/?q=ai:totur.umit"Çanak, İbrahim"https://www.zbmath.org/authors/?q=ai:canak.ibrahimSummary: Given a real-valued integrable function \(f(x, y, z)\) which is integrable over \([0,\infty)\times [0,\infty)\times [0,\infty)\), let \(s(x, y, z)\) denote its integral over \([0,x]\times [0,y]\times [0,z]\) and let \(\sigma (x,y,z)\) denote its \((C, 1, 1, 1)\) mean, the average of \(s(x, y, z)\) over \([0,x]\times [0,y]\times [0,z]\), where \(x,y,z >0.\) We give one-sided Tauberian conditions of Landau and Hardy type under which convergence of \(s(x, y, z)\) follows from \((C, 1, 1, 1)\) summability of \(s(x, y, z)\). We obtain convergence of \(s(x, y, z)\) from its \((C, 1, 1, 1)\) summability provided that \(s(x, y, z)\) is slowly oscillating in certain senses. Furthermore, we extend a Tauberian theorem given by \textit{F. Móricz} [Stud. Math. 138, No. 1, 41--52 (2000; Zbl 0949.40012)] for improper double integrals to improper triple integrals.Generalized limits and sequence of matrices.https://www.zbmath.org/1456.400102021-04-16T16:22:00+00:00"Özgüç, İ."https://www.zbmath.org/authors/?q=ai:ozguc.ilknur"Taş, E."https://www.zbmath.org/authors/?q=ai:tas.emre"Yurdakadim, T."https://www.zbmath.org/authors/?q=ai:yurdakadim.tugbaLet \(\mathcal{B}=(B_i)_{i\in \mathbb{N}}\) be a sequence of infinite matrices with non-negative entries which defines a regular summability method. Denote by \(m\) the space of all bounded real-valued sequences equipped with the supremum norm.
The authors introduce and study two types of linear functionals on \(m\) which enjoy certain properties, called \(S_{\mathcal{B}}\)-limits and \(\mathcal{B}\)-Banach limits.
For instance, it is proved that the set \(SL_{\mathcal{B}}\) of \(S_{\mathcal{B}}\)-limits and the set \(BL_{\mathcal{B}}\) of \(\mathcal{B}\)-Banach limits have a non-empty intersection and that \(BL_{\mathcal{B}}\) coincides with the set of ordinary Banach limits
if and only if \(\mathcal{B}\) is strongly regular.
Reviewer: Jan-David Hardtke (Leipzig)On statistically convergent sequences of closed sets.https://www.zbmath.org/1456.400062021-04-16T16:22:00+00:00"Talo, Özer"https://www.zbmath.org/authors/?q=ai:talo.ozer"Sever, Yurdal"https://www.zbmath.org/authors/?q=ai:sever.yurdal"Başar, Feyzi"https://www.zbmath.org/authors/?q=ai:basar.feyziSummary: In this paper, we give the definitions of statistical inner and outer limits for sequences of closed sets in metric spaces.
We investigate some properties of statistical inner and outer limits. For sequences of closed sets if its statistical outer and statistical inner limits coincide, we say that the sequence is Kuratowski statistically convergent. We prove some proporties for Kuratowski statistically convergent sequences. Also, we examine the relationship between Kuratowski statistical convergence and Hausdorff statistical convergence.Acceleration of convergence of some infinite sequences \(\{A_n\}\) whose asymptotic expansions involve fractional powers of \(n\) via the \(\widetilde{d}^{(m)}\) transformation.https://www.zbmath.org/1456.650022021-04-16T16:22:00+00:00"Sidi, Avram"https://www.zbmath.org/authors/?q=ai:sidi.avramSummary: In this paper, we discuss the application of the author's \(\tilde{d}^{(m)}\) transformation [Practical extrapolation methods. Theory and applications. Cambridge: Cambridge University Press (2003; Zbl 1041.65001), Section 6.6] to accelerate the convergence of infinite series \(\sum\nolimits^{\infty}_{n = 1} a_n\) when the terms \(a_n\) have asymptotic expansions that can be expressed in the form
\[
a_n \sim (n!)^{s/m} \exp \left[\sum\limits^m_{i=0} q_i n^{i/m}\right] \sum\limits^\infty_{i=0} w_i n^{\gamma-i/m} \quad\text{as } n \to \infty, \quad s \text{ integer.}
\]
We discuss the implementation of the \(\widetilde{d}^{(m)}\) transformation via the recursive W-algorithm of the author. We show how to apply this transformation and how to assess in a reliable way the accuracies of the approximations it produces, whether the series converge or they diverge. We classify the different cases that exhibit unique numerical stability issues in floating-point arithmetic. We show that the \(\widetilde{d}^{(m)}\) transformation can also be used efficiently to accelerate the convergence of infinite products \(\prod\nolimits^{\infty}_{n=1} (1 + v_n)\), where \(v_n \sim \sum\nolimits^{\infty}_{i=0} e_i n^{-t/m - i/m}\) as \(n \to \infty\), \(t \geq m + 1\) an integer. Finally, we give several numerical examples that attest the high efficiency of the \(\widetilde{d}^{(m)}\) transformation for the different cases.Approximation by sub-matrix means of multiple Fourier series in the Hölder metric.https://www.zbmath.org/1456.420102021-04-16T16:22:00+00:00"Krasniqi, Xhevat Z."https://www.zbmath.org/authors/?q=ai:krasniqi.xhevat-zahir|krasniqi.xhevat-zSummary: In this paper some results on approximation by sub-matrix means of multiple Fourier series in the Hölder metric are obtained. Our results are applicable for a wider class of sequences and give a better degree of approximation than those presented previously by others.A continuant and an estimate of the remainder of the interpolating continued \(C\)-fraction.https://www.zbmath.org/1456.300082021-04-16T16:22:00+00:00"Pahirya, M. M."https://www.zbmath.org/authors/?q=ai:pahirya.mykhaylo-m|pagirya.m-mSummary: The problem of the interpolation of functions of a real variable by interpolating continued \(C\)-fraction is investigated. The relationship between the continued fraction and the continuant was used. The properties of the continuant are established. The formula for the remainder of the interpolating continued \(C\)-fraction proved. The remainder expressed in terms of derivatives of the functional continent. An estimate of the remainder was obtained. The main result of this paper is contained in the following Theorem 5:
Let \(\mathcal{R}\subset \mathbb{R}\) be a compact, \(f \in \mathbf{C}^{(n+1)}(\mathcal{R})\) and the interpolating continued \(C\)-fraction (\(C\)-ICF) of the form
\[D_n(x)=\frac{P_n(x)}{Q_n(x)}=a_0+\frac{K}{k=1}{n}\frac{a_k(x-x_{k-1})}{1}, a_k \in \mathbb{R}, \; k=\overline{0,n},\]
be constructed by the values the function \(f\) at nodes \(X=\{x_i : x_i \in \mathcal{R}\), \(x_i\neq x_j\), \(i\neq j\), \(i,j=\overline{0,n}\}\). If the partial numerators of \(C\)-ICF satisfy the condition of the Paydon-Wall type, that is \(0<a^* \operatorname{diam} \mathcal{R} \leq p\), then
\[\begin{aligned} |f(x)-D_n(x)|\leq \frac{f^*\prod\limits_{k=0}^n |x-x_k|}{(n+1)! \Omega_n(t)} \Bigg( \kappa_{n+1}(p)+\sum_{k=1}^r \binom{n+1}{k} (a^*)^k \sum_{i_1=1}^{n+1-2k} \kappa_{i_1}(p)\times \\
\times \sum_{i_2=i_1+2}^{n+3-3k} \kappa_{i_2-i_1-1}(p)\dots \sum_{i_{k-1}=i_{k-2}+2}^{n-3} \kappa_{i_{k-1}-i_{k-2}-1}(p) \sum_{i_k=i_{k-1}+2}^{n-1} \kappa_{i_k-i_{k-1}-1}(p) \kappa_{n-i_k}(p)\Bigg),\end{aligned}\]
where \(f^*= \max\limits_{0\leq m \leq r}\max\limits_{x \in \mathcal{R}} |f^{(n+1-m)}(x)|\), \( \kappa_n(p)=\frac{(1+\sqrt{1+4p})^n-(1-\sqrt{1+4p})^n}{2^n \sqrt{1+4p}}\), \(a^*=\max\limits_{2\leqslant i \leqslant n}|a_i|\), \(p=t(1-t)\), \(t\in(0;\frac{1}{2}]\), \(r=\big[\frac{n}{2}\big]\).A new type of difference class of interval numbers.https://www.zbmath.org/1456.460032021-04-16T16:22:00+00:00"Baruah, Achyutananda"https://www.zbmath.org/authors/?q=ai:baruah.achyutananda"Dutta, Amar Jyoti"https://www.zbmath.org/authors/?q=ai:dutta.amar-jyotiSummary: In this article we introduce the notation difference operator \(\Delta_m\) (\(m \geq 0\) be an integer) for studying some properties defined with interval numbers. We introduced the classes of sequence \(\bar{\ell}(p)(\Delta_m)\), \(\bar{c}(p)(\Delta_m)\) and \(\bar{c}_0(\Delta_m)\) and investigate different algebraic properties like completeness, solidness, convergence free etc.On a new class of generalized difference sequence spaces of fractional order defined by modulus function.https://www.zbmath.org/1456.460082021-04-16T16:22:00+00:00"Yaying, T."https://www.zbmath.org/authors/?q=ai:yaying.tajaSummary: \textit{P. Baliarsingh} and \textit{S. Dutta} [Bol. Soc. Parana. Mat. (3) 33, No. 1, 49--57 (2015; Zbl 1412.46006); Appl. Math. Comput. 250, 665--674 (2015; Zbl 1328.46002)] introduced the fractional difference operator \(\Delta^\alpha \), defined by \(\Delta^\alpha (x_k) =\sum^\alpha_{i=0} (-1)^i \frac{\Gamma(\alpha+1)}{i!\Gamma(\alpha-i+1)} x_{k+i}\) and defined new classes of generalized difference sequence spaces of fractional order \(X ( \Gamma, \Delta^\alpha, u)\) where \(X = \{ \ell_\infty, c, c_0\}\). Moreover, \textit{U. Kadak} [Int. J. Math. Math. Sci. 2015, Article ID 984283, 6 p. (2015; Zbl 06873276)] studied strongly Cesàro and statistical difference sequence space of fractional order involving lacunary sequences using the fractional difference operator \(\Delta^\alpha_v\) defined by \(\Delta^\alpha_v (x_k) = \sum^\alpha_{i=0} (-1)^i \frac{\Gamma(\alpha+1)}{i!\Gamma(\alpha-i+1)} v_k+{}_i x_k+ i\), where \(v=(v_k)\) is any fixed sequence of positive real or complex numbers.
Following Baliarsingh and Dutta and Kadak, we introduce paranormed difference sequence spaces \( N_\theta (\Delta^\alpha_v , f, p)\) and \(S_\theta (\Delta^\alpha_v,f,p)\) of fractional order involving a lacunary sequence \(\theta\) and a modulus function \(f\). We investigate topological structures of these spaces and examine various inclusion relations.Some new triple sequence spaces over \(n\)-normed space.https://www.zbmath.org/1456.460072021-04-16T16:22:00+00:00"Jalal, Tanweer"https://www.zbmath.org/authors/?q=ai:jalal.tanweer"Malik, Ishfaq Ahmad"https://www.zbmath.org/authors/?q=ai:malik.ishfaq-ahmadSummary: Triple sequence spaces were introduced by \textit{A. Şahiner} et al. [Selçuk J. Appl. Math. 8, No. 2, 49--55 (2007; Zbl 1152.40306)]. The main objective of this paper is to define some new classes of triple sequences over $n$-normed space by means of Musielak-Orlicz function and difference operators. We also study some algebraic and topological properties of these new sequence spaces.On some double sequence spaces of interval number.https://www.zbmath.org/1456.460062021-04-16T16:22:00+00:00"Gölbol, Sibel Yasemin"https://www.zbmath.org/authors/?q=ai:golbol.sibel-yasemin"Esi, Ayhan"https://www.zbmath.org/authors/?q=ai:esi.ayhan"Değer, Uğur"https://www.zbmath.org/authors/?q=ai:deger.ugurSummary: \textit{A. Esi} and \textit{S. Yasemin} [``Some spaces of sequences of interval numbers defined by a modulus function'', Global J. Math. Anal. 2, No. 1, 11--16 (2014; \url{doi:10.14419/gjma.v2i1.2005})] defined the metric spaces \(\overline{c}_0(f,p,s)\), \(\overline{c}(f,p,s)\), \(\overline{l}_\infty (f,p,s)\) and \(\overline{l}_p(f,p,s)\) of sequences of interval numbers by a modulus function. In this study, we consider a generalization for double sequences of these metric spaces by taking a \(\psi\) function, satisfying the following conditions, instead of \(s\) parameter. For this aim, let \(\psi(k, l)\) be a positive function for all \(k, l \in \mathbb{N}\) such that
\begin{itemize}
\item[(i)] \(\underset{k,l\rightarrow\infty}{\lim} \psi(k,l)=0,\)
\item[(ii)] \(\Delta_2 \psi(k,l) = \psi(k-1,l-1) -2\psi(k,l) + \psi(k+1,l+1) \geq 0\)
\end{itemize}
or
\(\psi(k,l)=1\).
Therefore, according to class of functions which satisfying the conditions (i) and (ii) we deal with the metric spaces \(\overline{c}_0^2(f,p,\psi)\), \(\overline{c}^2(f,p,\psi)\), \(\overline{l}_\infty^2(f,p,\psi)\) and \(\overline{l}_p^2(f,p,\psi)\) of double sequences of interval numbers defined by a modulus function.Infinite matrices and some matrix transformations.https://www.zbmath.org/1456.400072021-04-16T16:22:00+00:00"Eren, Rahmet Savaş"https://www.zbmath.org/authors/?q=ai:eren.rahmet-savasSummary: The goal of this paper is to define the spaces $V_{\sigma_ 0}^\lambda (p)$ and $V_{\sigma}^\lambda (p) $ by using de la Vallée Poussin and invariant mean. Furthermore we
characterize certain matrices in $V_\sigma^\lambda$ which will fill a gap in the existing
literature.Generalized weighted statistical convergence of double sequences and applications.https://www.zbmath.org/1456.400032021-04-16T16:22:00+00:00"Cinar, Muhammed"https://www.zbmath.org/authors/?q=ai:cinar.muhammed"Et, Mikail"https://www.zbmath.org/authors/?q=ai:et.mikailSummary: In this paper we introduce the concept generalized weighted statistical convergence of double sequences. Some relations between weighted $(\lambda,\mu)$-statistical convergence and strong $(\overline N_{\lambda,\mu},p,q,\alpha,\beta)$-summablity of double sequences are examined. Furthermore we apply our new summability method to prove a Korovkin type theorem.Almost everywhere strong \(C,1,0\) summability of 2-dimensional trigonometric Fourier series.https://www.zbmath.org/1456.420112021-04-16T16:22:00+00:00"Goginava, Ushangi"https://www.zbmath.org/authors/?q=ai:goginava.ushangiSummary: In this paper we study the a. e. exponential strong \((C, 1, 0)\) summability of of the 2-dimensional trigonometric Fourier series of the functions belonging to \(L (\log^+L)^2\).Fuzzy normed linear sequence space.https://www.zbmath.org/1456.460052021-04-16T16:22:00+00:00"Das, Paritosh Chandra"https://www.zbmath.org/authors/?q=ai:das.paritosh-chandraSummary: In this article we introduce the notion of class of sequences \(bv_p^F(X)$, $1 \leq p < \infty\) with the concept of fuzzy norm. We study some of its properties such as completeness, solidness, symmetricity and convergence free. Also, we establish some inclusion results.On compact operators on some sequence spaces related to matrix \(B(r,s,t)\).https://www.zbmath.org/1456.470102021-04-16T16:22:00+00:00"Demiriz, Serkan"https://www.zbmath.org/authors/?q=ai:demiriz.serkan"Kara, Emrah Evren"https://www.zbmath.org/authors/?q=ai:kara.emrah-evrenSummary: In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the spaces \(c_0(B)\), \(\ell_\infty(B)\) and \(\ell_{p}(B)\) which have recently been introduced in [\textit{A. Sönmez}, Comput. Math. Appl. 62, No. 2, 641--650 (2011; Zbl 1228.40006)]. Further, by using the Hausdorff measure of noncompactness, we characterize some classes of compact operators on these spaces.Wijsman lacunary ideal invariant convergence of double sequences of sets.https://www.zbmath.org/1456.400042021-04-16T16:22:00+00:00"Dündar, Erdinc"https://www.zbmath.org/authors/?q=ai:dundar.erdinc"Akin, Nimet Pancaroğlu"https://www.zbmath.org/authors/?q=ai:akin.nimet-pancarogluSummary: In this paper, we study the concepts of Wijsman lacunary invariant convergence, Wijsman lacunary invariant statistical convergence, Wijsman lacunary \({\mathcal{I}}_2\)-invariant convergence \(({\mathcal{I}}^{{\sigma}{\theta}}_{W_2} )\), Wijsman lacunary \({\mathcal{I}}^\ast_2\)-invariant convergence \(({\mathcal{I}}^{\ast} {}^{{\sigma}{\theta}}_{W_2} )\), Wijsman \(p\)-strongly lacunary invariant convergence \(([W_2 N_{\sigma \theta}]_p)\) of double sequence of sets and investigate the relationships among Wijsman lacunary invariant convergence, \([W_2 N_{\sigma \theta}]_p, {\mathcal{I}}^{{\sigma}{\theta}}_{W_2}\) and \({\mathcal{I}}^{\ast} {}^{{\sigma}{\theta}}_{W_2} \). Also, we introduce the concepts of \({\mathcal{I}}^{{\sigma}{\theta}}_{W_2} \)-Cauchy double sequence and \({\mathcal{I}}^{\ast} {}^{{\sigma}{\theta}}_{W_2} \)-Cauchy double sequence of sets.\(\mathcal{I}_2\)-uniform convergence of double sequences of functions.https://www.zbmath.org/1456.400022021-04-16T16:22:00+00:00"Dündar, Erdinç"https://www.zbmath.org/authors/?q=ai:dundar.erdinc"Altayb, Bilal"https://www.zbmath.org/authors/?q=ai:altayb.bilalSummary: In this work, we discuss various kinds of $\mathcal{I}_2$-uniform convergence for double sequences of functions and introduce the concepts of $\mathcal{I}_2$ and $\mathcal{I}_2^*$-uniform convergence, $\mathcal{I}_2$ and $\mathcal{I}_2^*$-uniformly Cauchy sequences for double sequences of
functions. Then, we show the relation between them.Factorization of $C$-finite sequences.https://www.zbmath.org/1456.682312021-04-16T16:22:00+00:00"Kauers, Manuel"https://www.zbmath.org/authors/?q=ai:kauers.manuel"Zeilberger, Doron"https://www.zbmath.org/authors/?q=ai:zeilberger.doronSummary: We discuss how to decide whether a given $C$-finite sequence can be written nontrivially as a product of two other $C$-finite sequences.
For the entire collection see [Zbl 1391.33001].Limit points of folding sequences.https://www.zbmath.org/1456.400012021-04-16T16:22:00+00:00"Sequin, Matt"https://www.zbmath.org/authors/?q=ai:sequin.mattThe author introduces folding sequences and discusses some of their properties focusing on limit points.
A folding sequence is a sequence $(f_n)$ in $[0,1]$ such that $f_1=1/2$ and for every $n>1$ either $f_n=f_{n-1}/2$ or $f_n=(f_{n-1}+1)/2$.Strongly statistical convergence.https://www.zbmath.org/1456.400052021-04-16T16:22:00+00:00"Kaya, U."https://www.zbmath.org/authors/?q=ai:kaya.ufuk|kaya.utku"Aral, N. D."https://www.zbmath.org/authors/?q=ai:aral.nazlim-denizSummary: We introduce a concept of \(A\)-strongly statistical convergence for sequences of complex numbers, where \(A = (a_{ nk })_{ n,k \in \mathbb{N} }\) is an infinite matrix with nonnegative entries. A sequence \((x_n)\) is called strongly convergent to \(L\) if \(\lim_{n\to \infty }{\sum}_{k=1}^{\infty } a_{nk} |x_k-L|=0\), in the ordinary sense. In the definition of \(A\)-strongly statistical limit, we use the statistical limit instead of the ordinary limit with a convenient density. We study some densities and show that the \((a_{ nk } )\)-strongly statistical limit is an \(( {a}_{m_nk} )\)-strong limit, where the density of the set \(\{m_n \in \mathbb{N} : n \in \mathbb{N} \}\) is equal to 1. We introduce the notion of dense positivity for nonnegative sequences. A nonnegative sequence \((r_n)\) is dense positive provided the limit superior of a subsequence \(( r_{m_n} )\) is positive for all \((m_n)\) with density equal to 1. We show that the dense positivity of \((r_n)\) is a necessary and sufficient condition for the uniqueness of \(A\)-strongly statistical limit, where \(A = (a_{ nk } )\) and \(r_n = {\sum}_{k=1}^{\infty }{a}_{nk} \). Furthermore, necessary conditions for the regularity, linearity and multiplicativity of the \(A\)-strongly statistical limit are established.