Recent zbMATH articles in MSC 40 https://zbmath.org/atom/cc/40 2023-09-22T14:21:46.120933Z Werkzeug A special form of slower divergent series https://zbmath.org/1517.40001 2023-09-22T14:21:46.120933Z "Stoica, George" https://zbmath.org/authors/?q=ai:stoica.george "Wardat, Yousef" https://zbmath.org/authors/?q=ai:wardat.yousef From the text: J. Marshall Ash [\textit{J.~M.~Ash}, Neither a worst convergent series nor a best divergent series exists, Coll. Math. J. 28, No.~4, 296--297 (1997, Zbl 1516.40001)] gave a simple and elegant proof to the fact, first noticed by \textit{N. H. Abel} [J. Reine Angew. Math. 3, 79--82 (1828; ERAM 003.0093cj)], that for each divergent positive series there is always another one which diverges slower. We shall prove below that the general term of the slower divergent series can always be chosen as a universal power function of the general term of the original series. On statistical deferred $$A$$-convergence of uncertain sequences https://zbmath.org/1517.40002 2023-09-22T14:21:46.120933Z "Baliarsingh, P." https://zbmath.org/authors/?q=ai:baliarsingh.pinakadhar Summary: In this paper, as a part of uncertainty theory, we discuss various concepts of convergence and statistical convergence of complex uncertain sequences. Using uncertain variables, the convergence of uncertain sequences of complex numbers such as the idea of convergence in measure, convergence in mean, convergence in distribution and convergence uniformly almost surely via deferred Cesàro mean and a regular matrix $$A$$ are introduced. Also, some relations among these convergence are established. Certain results on deferred statistical convergence of order $$\alpha$$, $$(0 < \alpha \leq 1)$$ for complex uncertain sequences are studied in detail. Further remarks on uniform statistical convergence of order $$\alpha$$ https://zbmath.org/1517.40003 2023-09-22T14:21:46.120933Z "Pal, Sudip Kumar" https://zbmath.org/authors/?q=ai:pal.sudip-kumar Following a summability method (called uniform statistical convergence by \textit{H. Albayrak} and \textit{S. Pehlivan} [Appl. Math. Lett. 23, No. 10, 1203--1207 (2010; Zbl 1206.40001)] corresponding to uniform density by \textit{V. Baláž} and \textit{T. Šalát} [Math. Commun. 11, No. 1, 1--7 (2006; Zbl 1114.40001)], the author introduces a new concept called $$I_u$$ convergence of order $$\alpha$$, the uniform statistical convergence of order $$\alpha$$, $$0<\alpha\leq 1$$, and obtains its basic properties. It is worth noting that the author states some open problems for interested readers. Reviewer: İbrahim Çanak (İzmir) Applications of statistical convergence in complex uncertain sequences via deferred Riesz mean https://zbmath.org/1517.40004 2023-09-22T14:21:46.120933Z "Saini, Kavita" https://zbmath.org/authors/?q=ai:saini.kavita "Raj, Kuldip" https://zbmath.org/authors/?q=ai:raj.kuldip Summary: In this paper, we intend to make a new approach to introduce the concept of $$t_n$$-statistical convergence of complex uncertain sequences like $$t_n$$-statistical convergence, almost surely (a.s.), $$t_n$$-statistical convergence in measure, $$t_n$$-statistical convergence in mean, $$t_n$$-statistical convergence in distribution and $$t_n$$-statistical convergence in uniformly almost surely (u.a.s). Various inclusion relations are established between newly formed sequence spaces. Failure of approximation of odd functions by odd polynomials https://zbmath.org/1517.41002 2023-09-22T14:21:46.120933Z "Mashreghi, Javad" https://zbmath.org/authors/?q=ai:mashreghi.javad "Parisé, Pierre-Olivier" https://zbmath.org/authors/?q=ai:parise.pierre-olivier "Ransford, Thomas" https://zbmath.org/authors/?q=ai:ransford.thomas-j Summary: We construct a Hilbert holomorphic function space $$H$$ on the unit disk such that the polynomials are dense in $$H$$, but the odd polynomials are not dense in the odd functions in $$H$$. As a consequence, there exists a function $$f$$ in $$H$$ that lies outside the closed linear span of its Taylor partial sums $$s_n(f)$$, so it cannot be approximated by any triangular summability method applied to the $$s_n(f)$$. We also show that there exists a function $$f$$ in $$H$$ that lies outside the closed linear span of its radial dilates $$f_r$$, $$r<1$$. Multivariate approximation in $$\varphi$$-variation for nonlinear integral operators via summability methods https://zbmath.org/1517.41007 2023-09-22T14:21:46.120933Z "Aslan, İsmail" https://zbmath.org/authors/?q=ai:aslan.ismail|aslan.ismail.1 Summary: We consider convolution-type nonlinear integral operators endowed with Musielak-Orlicz $$\varphi$$-variation. Our aim is to get more powerful approximation results with the help of summability methods. In this study, we use $$\varphi$$-absolutely continuous functions for our convergence results. Moreover, we study the order of approximation using suitable Lipschitz class of continuous functions. A general characterization theorem for $$\varphi$$-absolutely continuous functions is also obtained. We also give some examples of kernels in order to verify our approximations. At the end, we indicate our approximations in figures together with some numerical computations. The spectrum and fine spectrum of generalized Rhaly-Cesàro matrices on $$c_0\ \mathrm{and}\ c$$ https://zbmath.org/1517.47059 2023-09-22T14:21:46.120933Z "Yildirim, Mustafa" https://zbmath.org/authors/?q=ai:yildirim.mustafa "Mursaleen, Mohammad" https://zbmath.org/authors/?q=ai:mursaleen.mohammad "Doğan, Çağla" https://zbmath.org/authors/?q=ai:dogan.cagla Summary: The generalized Rhaly Cesàro matrices $$A_{\alpha}$$ are the triangular matrix with nonzero entries $$a_{nk} = \alpha^{n-k}/(n + 1)$$ with $$\alpha \in [0,1]$$. In [Proc. Am. Math. Soc. 86, 405--409 (1982; Zbl 0505.47021)], \textit{H. C. Rhaly jun.}\ determined boundedness, compactness of generalized Rhaly Cesàro matrices on $$\ell_2$$ Hilbert space and showed that its spectrum is $$\sigma(A_{\alpha},\ell_2) = \{1/n\} \cup \{0\}$$. Also, in [Linear Multilinear Algebra 26, No. 1--2, 49--58 (1990; Zbl 0697.15009)], lower bounds for these classes were obtained under certain restrictions on $$\ell_p$$ by \textit{B. E. Rhoades}. In the present paper, boundedness, compactness, spectra, the fine spectra and subdivisions of the spectra of generalized Rhaly Cesàro operator on $$c_0\ \mathrm{and}\ c$$ have been determined. Reply to: Comment on: `On the characteristic polynomial of an effective Hamiltonian{'}'' https://zbmath.org/1517.81053 2023-09-22T14:21:46.120933Z "Zheng, Yong" https://zbmath.org/authors/?q=ai:zheng.yong Summary: In a recent comment by \textit{F. M. Fernández} [Phys. Lett., A 452, Article ID 128456, 3 p. (2022; Zbl 1515.81103)], it has been argued that our solution method of an effective Hamiltonian based on the characteristic polynomial [the author, Phys. Lett., A 443, Article ID 128215, 5 p. (2022; Zbl 1498.81077)] had been developed several years earlier by \textit{L. E. Fried} and \textit{G. S. Ezra} [J. Chem. Phys. 90, 6378--6390 (1989; \url{doi:10.1063/1.456303})]. We show here several important differences between our treatment and the resummation method proposed previously by Fried and Ezra.