Recent zbMATH articles in MSC 40https://zbmath.org/atom/cc/402021-01-08T12:24:00+00:00Werkzeug\({A^I}\)-statistical convergence and strong \({A^I}\)-convergence of sequences of fuzzy numbers with respect to the Orlicz function.https://zbmath.org/1449.400022021-01-08T12:24:00+00:00"Feng, Xue"https://zbmath.org/authors/?q=ai:feng.xue"Gong, Zengtai"https://zbmath.org/authors/?q=ai:gong.zengtaiSummary: As an extension of the ideal statistical convergent sequence space of fuzzy number, based on Orlicz functions and a non-negative regular matrix \(A = \{a_{nk}\}\), we defined and discussed \({A^I}\)-statistical convergence and strong \({A^I}\)-convergence of sequences of fuzzy numbers. In addition, the relationship of the two different convergences was investigated. If a sequence of fuzzy number is strongly \({A^I}\)-convergent then it is \({A^I}\)-statistically convergent.Cesàro-like operators.https://zbmath.org/1449.470472021-01-08T12:24:00+00:00"Rhoades, B. E."https://zbmath.org/authors/?q=ai:rhoades.billy-e"Trutt, D."https://zbmath.org/authors/?q=ai:trutt.dSummary: In [\textit{B. K. Ghosh} et al., Proc. Am. Math. Soc. 66, 261--265 (1977; Zbl 0386.47009)], it was shown that the lower triangular generalized Hausdorff matrix \(H_\alpha\) with nonzero entries \(h_{nk}= (n+ \alpha+1)^{-1}\), for \(\alpha \geq 0\), is subnormal on \(\ell^2\) if and only if \(\alpha =0,1,2,\dots\). For \(0<h \leq 1\), the weighted Cesàro operator \(C'_h:\{a_n\} \rightarrow \{b_n\}\) on \(\ell^2\), when \(b_n= (a_0+d_1a_1+ \cdots +d_na_n)/(n+1)d_n\), is subnormal when \(d^2_j= \Gamma (j+1) \Gamma (h)/ \Gamma (j+h)\).
In this paper, we show that, when \(d_j= \Gamma (j+1) \Gamma (h)/ \Gamma (j+h)\), the square of the weights chosen above, then the corresponding operator \(C_h\) is bounded on \(\ell^2\) for \(0<h<3/2\), that \(H_\alpha\) is bounded on \(\ell^2\) for all non-integer \(\alpha <0\), and that \(C_h\) is closely related to \(H_{h-1}\). This relationship leads to our main result that \(C_h\) is only subnormal when \(h=1\), when it corresponds to the original Cesàro operator with \(\alpha =0\) and each \(d_j=1\).A note on farthest point problem in Banach spaces.https://zbmath.org/1449.460192021-01-08T12:24:00+00:00"Som, Sumit"https://zbmath.org/authors/?q=ai:som.sumit"Savas, Ekrem"https://zbmath.org/authors/?q=ai:savas.ekremSummary: Farthest point problem states that ``Must every uniquely remotal set in a Banach space be singleton?'' In this paper we introduce the notion of partial ideal statistical continuity of a function which is way weaker than continuity of a function. We give an example to show that partial ideal statistical continuity is weaker than continuity. In this paper we use ideal summability to give some answers to FPP problem which improves some former results. We prove that if \(E\) is a non-empty, bounded, uniquely remotal subset in a real Banach space \(X\) such that \(E\) has a Chebyshev center \(c\) and the farthest point map \(F:X\rightarrow E\) restricted to \([c,F(c)]\) is partially ideal statistically continuous at \(c\) then \(E\) is singleton.Cesàro means of subsequences of double sequences.https://zbmath.org/1449.400092021-01-08T12:24:00+00:00"Taş, Emre"https://zbmath.org/authors/?q=ai:tas.emre"Orhan, Cihan"https://zbmath.org/authors/?q=ai:orhan.cihanSummary: In this paper we characterize the convergence and \((C,1,1)\) summability of a double sequence. In particular we study conditions under which the convergence or \((C,1,1)\) summability of a double sequence carry over to that of its subsequences, and conversely, whether these properties for suitable subsequences imply them for the sequence itself. We show, for instance, that a bounded double sequence is \((C,1,1)\) summable if and only if almost all of its subsequences are \((C,1,1)\) summable.Product of statistical probability convergence and its applications to Korovkin-type theorem.https://zbmath.org/1449.400032021-01-08T12:24:00+00:00"Jena, Bidu Bhusan"https://zbmath.org/authors/?q=ai:jena.bidu-bhusan"Paikray, Susanta Kumar"https://zbmath.org/authors/?q=ai:paikray.susanta-kumarSummary: In the present work, we introduce and study the notion of statistical probability convergence for sequences of random variables as well as the concept of statistical convergence for sequences of real numbers, which are defined over a Banach space via product of deferred Cesàro and deferred Nörlund summability means. We first establish a theorem presenting a connection between them. Based upon our proposed method, we then prove a Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results (in classical as well as statistical versions). Finally, an illustrative example is presented here by means of the generalized Meyer-König and Zeller operators for a positive sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.A bounds Tauberian theorem.https://zbmath.org/1449.400102021-01-08T12:24:00+00:00"Stenger, Allen"https://zbmath.org/authors/?q=ai:stenger.allenSummary: We weaken the hypothesis and the conclusion of a Hardy-Littlewood Tauberian theorem, and apply the new theorem to deduce asymptotic behavior of the coefficients of an exponentiated lacunary series.Fundamental theorems of summability theory for a new type of subsequences of double sequences.https://zbmath.org/1449.400072021-01-08T12:24:00+00:00"Dumitru, Raluca"https://zbmath.org/authors/?q=ai:dumitru.raluca"Franco, Jose A."https://zbmath.org/authors/?q=ai:franco.jose-aSummary: In 2000, the notion of a subsequence of a double sequence was introduced [\textit{R. F. Patterson}, Int. J. Math. Math. Sci. 23, No. 1, 1--9 (2000; Zbl 0954.40005)]. Using this definition, a multidimensional analogue to a result from H. Steinhaus, that states that for any regular matrix \(A\), there exists a sequence of zeros and ones that is not \(A\)-summable, was proved. Additionally, an analogue of a result of R. C. Buck that states that a sequence \(x\) is convergent if and only if there exists a regular matrix \(A\) that sums every subsequence of \(x\) was presented. However, this definition imposes a restrictive condition on the entries of the double sequence that can be considered for the subsequence. In this article, we introduce a less restrictive new definition of a subsequence. We denote them by \(\beta \)-subsequences of a double sequence and show that analogues to these two fundamental theorems of summability still hold for these new subsequences.Cesàro summable sequence spaces over the non-Newtonian complex field.https://zbmath.org/1449.460052021-01-08T12:24:00+00:00"Kadak, Uğur"https://zbmath.org/authors/?q=ai:kadak.ugurSummary: The spaces \(\omega_0^p\), \(\omega^p\), and \(\omega_{\infty}^p\) can be considered the sets of all sequences that are strongly summable to zero, strongly summable, and bounded, by the Cesàro method of order \(1\) with index \(p\). Here we define the sets of sequences which are related to strong Cesàro summability over the non-Newtonian complex field by using two generator functions. Also we determine the \(\beta\)-duals of the new spaces and characterize matrix transformations on them into the sets of \(\ast\)-bounded, \(\ast\)-convergent, and \(\ast\)-null sequences of non-Newtonian complex numbers.On generalized geometric difference of six dimensional rough ideal convergent of triple sequence defined by Musielak-Orlicz function.https://zbmath.org/1449.400012021-01-08T12:24:00+00:00"Esi, Ayhan"https://zbmath.org/authors/?q=ai:esi.ayhan"Subramanian, N."https://zbmath.org/authors/?q=ai:subramanian.nagarajanSummary: We introduce a rough ideal convergent of triple sequence spaces defined by Musielak-Orlicz function, using an six dimensional infinite matrix, and a generalized geometric difference Zweier six dimensional matrix operator \(B^p_{(abc)}\) of order \(p\). We obtain some topological and algebraic properties of these spaces.Development of \(N\)-multiple power series into \(N\)-dimensional regular \(C\)-fraction.https://zbmath.org/1449.300052021-01-08T12:24:00+00:00"Kuchminska, Kh. Yo."https://zbmath.org/authors/?q=ai:kuchminska.kh-yo"Vozna, S. M."https://zbmath.org/authors/?q=ai:vozna.s-mThe authors propose a possible algorithm for the expansion of a formal \(N\)-multiple power series into a functional \(N\)-dimensional regular \(C\)-fraction corresponding to this series. In this case, the algorithms used provide the expansion into the corresponding and corresponding two-dimensional fractions for the power and double power series.
Reviewer: L. N. Chernetskaja (Kyïv)Applications of Riesz mean and lacunary sequences to generate Banach spaces and AK-BK spaces.https://zbmath.org/1449.460072021-01-08T12:24:00+00:00"Raj, Kuldip"https://zbmath.org/authors/?q=ai:raj.kuldip"Esi, Ayhan"https://zbmath.org/authors/?q=ai:esi.ayhan"Pandoh, Suruchi"https://zbmath.org/authors/?q=ai:pandoh.suruchiSummary: In this paper we establish some wide-ranging spaces of sequences generated by Riesz mean associated with lacunary sequences and multiplier sequences of Orlicz function. We have encompassed some topological and algebraic properties of these sequence spaces. We also make an effort to prove that these spaces are Banach and AK-BK spaces. Finally, we prove that these sequence spaces are topologically isomorphic.On approximation of the rate of convergence of Fourier series in the generalized Hölder metric by deferred Nörlund mean.https://zbmath.org/1449.420062021-01-08T12:24:00+00:00"Pradhan, T."https://zbmath.org/authors/?q=ai:pradhan.tejaswini"Jena, B. B."https://zbmath.org/authors/?q=ai:jena.bidu-bhusan"Paikray, S. K."https://zbmath.org/authors/?q=ai:paikray.susanta-kumar"Dutta, H."https://zbmath.org/authors/?q=ai:dutta.hemen"Misra, U. K."https://zbmath.org/authors/?q=ai:misra.uma-kantaSummary: In this paper, we have studied an estimate of the rate of convergence of Fourier series in the generalized Hölder metric \(H_{L_r}^{(\omega)}\) space by using the deferred Nörlund mean and established some new results. Our results are more advanced that the already known and unify many other results available in the literature.Generalized functions asymptotically homogeneous along the unstable degenerated node.https://zbmath.org/1449.460322021-01-08T12:24:00+00:00"Drozhzhinov, Yuriĭ Nikolaevich"https://zbmath.org/authors/?q=ai:drozhzhinov.yu-n"Zav'yalov, Boris Ivanovich"https://zbmath.org/authors/?q=ai:zavialov.boris-ivanovichSummary: The generalized functions which have quasiasymptotics along the trajectories of one-parametric group are called asymptomatically homogeneous. The corresponding limit functions are homogeneous with respect to this group. In this paper we give the full description of asymptotically homogeneous generalized functions along the trajectories of unstable degenerated node. The obtained results are applied for description of homogeneous generalized functions for such trajectories in two dimensional case.Applications of four dimensional matrices to generating some roughly $I_2$-convergent double sequence spaces.https://zbmath.org/1449.460082021-01-08T12:24:00+00:00"Raj, Kuldip"https://zbmath.org/authors/?q=ai:raj.kuldip"Sharma, Charu"https://zbmath.org/authors/?q=ai:sharma.charuSummary: \textit{E. Dündar} [Numer. Funct. Anal. Optim. 37, No. 4, 480--491 (2016; Zbl 1365.40004)] introduced the notion of rough $I_2$-convergence and obtained two criteria for rough $I_2$-convergence. In the present paper we introduce some new rough ideal convergent sequence spaces of Musielak-Orlicz functions and four dimensional bounded-regular matrices. We study some topological and algebraic properties of these spaces. We also establish some inclusion relations between these spaces. Finally, we examine that these spaces are normal as well as monotone and sequence algebras.Certain properties of the sequence space \(\tilde \ell \left ({M,p,q} \right)\) of non-absolute type using four tuple band matrix \(B\left ({\tilde r,\tilde s,\tilde t,\tilde u} \right)\).https://zbmath.org/1449.460102021-01-08T12:24:00+00:00"Tripathy, Nilambar"https://zbmath.org/authors/?q=ai:tripathy.nilambar"Dutta, Salila"https://zbmath.org/authors/?q=ai:dutta.salilaSummary: In the present paper the sequence space \(\tilde l\left ({M, p, q} \right)\) of non-absolute type is introduced using an Orlicz function \(M\) along with a semi-norm \(q\) which is the domain of the generalized difference matrix \(B\left ({\tilde r, \tilde s, \tilde t, \tilde u} \right)\) and some topological as well as geometric properties of this space are obtained.Rate of convergence of wavelet series by Cesàro means.https://zbmath.org/1449.420512021-01-08T12:24:00+00:00"Ali, Mir Ahsan"https://zbmath.org/authors/?q=ai:ali.mir-ahsan"Sheikh, Neyaz A."https://zbmath.org/authors/?q=ai:sheikh.neyaz-ahmed|sheikh.neyaz-ahmadSummary: Wavelet frames have become a useful tool in time frequency analysis and image processing. Many authors worked in the field of wavelet frames and obtained various necessary and sufficient conditions. \textit{A. Ron} and \textit{Z. Shen} [J. Funct. Anal. 148, No. 2, 408--447 (1997; Zbl 0891.42018)] gave a charactarization of wavelet frames. \textit{J. J. Benedetto} and \textit{O. M. Treiber} [in: Wavelet transforms and time-frequency signal analysis. Boston, MA: Birkhäuser. 3--36 (2001; Zbl 1036.42032)], presented different works on the wavelet frames. Any function \(f\in L^2(R)\) can be expanded as an orthonormal wavelet series and pointwise convergence and uniform convergence of series have been discussed extensively by various authors [\textit{S. E. Kelly} et al., Bull. Am. Math. Soc., New Ser. 30, No. 1, 87--94 (1994; Zbl 0788.42014)]. In this paper we investigate the pointwise convergence of orthogonal wavelet series in Pringsheim's sense. Furthermore, we investigate Cesàro \(\vert C,1,1\vert\) summability and the strong Cesàro \(\vert C,1,1\vert\) summability of wavelet series.New interesting Euler sums.https://zbmath.org/1449.110872021-01-08T12:24:00+00:00"Nimbran, Amrik Singh"https://zbmath.org/authors/?q=ai:nimbran.amrik-singh"Sofo, Anthony"https://zbmath.org/authors/?q=ai:sofo.anthonySummary: We present here some new and interesting Euler sums obtained by means of related integrals and elementary approach. We supplement Euler's general recurrence formula with two general formulas of the form
\[\sum_{n\geqslant 1} O_n^{(m)}\left(\frac{1}{(2n -1)^p} + \frac {1}{(2n)^p}\right)\quad\text{and}\quad\sum_{n\geqslant 1}\frac{O_n}{(2n-1)^p(2n+1)^q}, \]
where \(\displaystyle O_n^{(m)}= \sum_{j=1}^n \frac{1}{(2j-1)^m}\). Two formulas for \(\zeta (5)\) are also derived.Exact evaluation of infinite series using double Laplace transform technique.https://zbmath.org/1449.400082021-01-08T12:24:00+00:00"Eltayeb, Hassan"https://zbmath.org/authors/?q=ai:eltayeb.hassan"Kılıçman, Adem"https://zbmath.org/authors/?q=ai:kilicman.adem"Mesloub, Said"https://zbmath.org/authors/?q=ai:mesloub.saidSummary: Double Laplace transform method was applied to evaluate the exact value of double infinite series. Further we generalize the current existing methods and provide some examples to illustrate and verify that the present method is a more general technique.On \(\mathcal{I}\)-covering mappings and 1-\(\mathcal{I}\)-covering mappings.https://zbmath.org/1449.400062021-01-08T12:24:00+00:00"Zhou, Xiangeng"https://zbmath.org/authors/?q=ai:zhou.xiangeng"Liu, Li"https://zbmath.org/authors/?q=ai:liu.li.3|liu.li.2|liu.li.6|liu.li|liu.li.4|liu.li.1|liu.li.7|liu.li.5Summary: In this paper, we introduce the concepts of \(\mathcal{I}\)-covering mappings and 1-\(\mathcal{I}\)-covering mappings, discuss the difference between sequence-covering and \(\mathcal{I}\)-covering mappings by some examples. With those concepts, we get some interesting properties of \(\mathcal{I}\)-covering (1-\(\mathcal{I}\)-covering) mappings and some characterizations of \(\mathcal{I}\)-covering (1-\(\mathcal{I}\)-covering) and compact mapping images of metric spaces.Bernstein Stancu operator of rough \(I\)-core of triple sequences.https://zbmath.org/1449.400052021-01-08T12:24:00+00:00"Subramanian, N."https://zbmath.org/authors/?q=ai:subramanian.nagarajan"Esi, A."https://zbmath.org/authors/?q=ai:esi.ayhan"Ozdemir, M. K."https://zbmath.org/authors/?q=ai:ozdemir.mustafa-kemalSummary: We introduce and study some basic properties of Bernstein-Stancu polynomials of rough $I$-convergent of triple sequences and also study the set of all Bernstein-Stancu polynomials of rough \(I\)-limits of a triple sequence and relation between analytic ness and Bernstein-Stancu polynomials of rough \(I\)-core of a triple sequence.Riesz triple almost lacunary \(\chi^3\) sequence spaces defined by a Orlicz function. I.https://zbmath.org/1449.460092021-01-08T12:24:00+00:00"Subramanian, N."https://zbmath.org/authors/?q=ai:subramanian.nagarajan"Esi, A."https://zbmath.org/authors/?q=ai:esi.ayhan"Aiyub, M."https://zbmath.org/authors/?q=ai:aiyub.mohammadSummary: In this paper we introduce a new concept for Riesz almost lacunary \({\chi}^3\) sequence spaces strong \(P\)-convergent to zero with respect to an Orlicz function and examine some properties of the resulting sequence spaces. We introduce and study statistical convergence of Riesz almost lacunary \({\chi}^3\) sequence spaces and some inclusion theorems are discussed.\(\mathcal{I}_2\)-asymptotically lacunary statistical equivalence of weight \(g\) of double sequences of sets.https://zbmath.org/1449.400042021-01-08T12:24:00+00:00"Kişi, Ömer"https://zbmath.org/authors/?q=ai:kisi.omerSummary: In this paper, our aim is to introduce new notions, namely, Wijsman asymptotically \(\mathcal{I}_2\)-statistical equivalence of weight \(g\), Wijsman strongly asymptotically \(\mathcal{I}_2\)-lacunary equivalence of weight \(g\) and Wijsman asymptotically \(\mathcal{I}_2\)-lacunary statistical equivalence of weight \(g\) of double set sequences. We mainly investigate their relationship and also make some observations about these classes.