Recent zbMATH articles in MSC 30Bhttps://www.zbmath.org/atom/cc/30B2021-04-16T16:22:00+00:00WerkzeugOn one Leontiev-Levin theorem.https://www.zbmath.org/1456.300492021-04-16T16:22:00+00:00"Krivosheev, Aleksandr Sergeevich"https://www.zbmath.org/authors/?q=ai:krivosheev.aleksandr-sergeevich"Kuzhaev, Arsen Fanilevich"https://www.zbmath.org/authors/?q=ai:kuzhaev.arsen-fanilevichSummary: In this work we study the relations between different densities of a positive sequence and related quantities. More precisely, in the work we consider the upper density, the maximal density introduced by G. Polya, the logarithmic block density, which seems to be introduced first by L. A. Rubel. In particular, there were obtained relations between the maximal density and a quantity being very close to the logarithmic block density. The results of these studies are applied for generalizing the classical statement obtained independently by A. F. Leont'ev B. and Ya. Levin on the completeness in a convex domain of a system of exponential monomials with positive exponents; we generalized this statement for the exponents with no density. We find out that for the aforementioned result, one can weaken the condition of the measurability of the sequence (that is, the existence of a density) and replace it by the identity of upper and maximal densities. Namely, we obtain a condition under which there holds the criterion of the completeness of the system of exponential monomials in convex domains. It should be noted that this criterion holds in a rather wide class of convex domains, for instance, having vertical and horizontal symmetry axes. The main role in solving this issues was played by the results of the studies by L. A. Rubel and P. Malliavin on relation between the growth of an entire function of exponential type along the imaginary axis and the logarithmic block density of its positive zeroes. These results were applied by these authors for studying the completeness of the system of exponentials in a horizontal strip.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 \(f^*= \max\limits_{0\leq m \leq r}\max\limits_{x \in \mathcal{R}} |f^{(n+1-m)}(x)|\), \( \kappa_n(p)=\frac{(1+\sqrt{1+4p})^n-(1-\sqrt{1+4p})^n}{2^n \sqrt{1+4p}}\), \(a^*=\max\limits_{2\leqslant i \leqslant n}|a_i|\), \(p=t(1-t)\), \(t\in(0;\frac{1}{2}]\), \(r=\big[\frac{n}{2}\big]\).Pseudostarlike and pseudoconvex Dirichlet series of order \(\alpha\) and type \( \beta \).https://www.zbmath.org/1456.300052021-04-16T16:22:00+00:00"Sheremeta, M. M."https://www.zbmath.org/authors/?q=ai:sheremeta.myroslav-mSummary: The concepts of the pseudostarlikeness of order \(\alpha\in [0,1)\) and type \(\beta\in (0,1]\) and the pseudoconvexity of order \(\alpha\) and type \(\beta\) are introduced for Dirichlet series with null abscissa of absolute convergence. In terms of coefficients, the pseudostarlikeness and the pseudoconvexity criteria of order \(\alpha\) and type \(\beta\) are proved. Let \(h\ge 1\), \(\Lambda=(\lambda_k)\) be an increasing to \(+\infty\) sequence of positive numbers \(( \lambda_1>h\). We call a conformal function of the form \(F(s)=e^{sh}+\sum\nolimits_{k=1}^{\infty}f_k\exp\{s\lambda_k\}\), \(s=\sigma+it\), in \(\Pi_0=\{s:\operatorname{Re}s<0\}\) pseudostarlike of order \(\alpha\in [0,\,1)\) and type \(\beta \in (0,\,1]\) if
\[\left|\frac{F'(s)}{F(s)}-h\right|<\beta\left|\frac{F'(s)}{F(s)}-(2\alpha-h)\right|,\quad s\in \Pi_0.\]
The main results of the article are contained in Theorems 1 and 2. Theorem 1 states: If \(\alpha \in [0, 1)\) and \(\beta \in (0, 1]\) such that \[\sum\limits_{k=1}^{\infty}\{(1+\beta)\lambda_k -2\beta\alpha -h(1-\beta)\}|f_k|\le 2\beta (h-\alpha) \]
then the function \(F\) is pseudostarlike of order \(\alpha\) and type \(\beta \). The corresponding results for Hadamard compositions of such series are also established.Complete characterization of bounded composition operators on the general weighted Hilbert spaces of entire Dirichlet series.https://www.zbmath.org/1456.300032021-04-16T16:22:00+00:00"Doan, Minh Luan"https://www.zbmath.org/authors/?q=ai:doan.minh-luan"Lê, Hai Khôi"https://www.zbmath.org/authors/?q=ai:le.hai-khoiSummary: We establish necessary and sufficient conditions for boundedness of composition operators on the most general class of Hilbert spaces of entire Dirichlet series with real frequencies. Depending on whether or not the space being considered contains any nonzero constant function, different criteria for boundedness are developed. Thus, we complete the characterization of bounded composition operators on all known Hilbert spaces of entire Dirichlet series of one variable.Truncation error bounds for the branched continued fraction \(\sum_{i_{1=1}}^N\frac{a_{i(1)}}{1}+\sum_{i_{2=1}}^{i_1}\frac{a_{i(2)}}{1}+\sum_{i_{3=1}}^{i_2}\frac{a_{i(3)}}{1}+\cdots \).https://www.zbmath.org/1456.300062021-04-16T16:22:00+00:00"Antonova, T. M."https://www.zbmath.org/authors/?q=ai:antonova.t-m"Dmytryshyn, R. I."https://www.zbmath.org/authors/?q=ai:dmytryshyn.r-iSummary: We analyze the problem of estimation of the error of approximation of a branched continued fraction, which is a multidimensional generalization of a continued fraction. By the method of fundamental inequalities, we establish truncation error bounds for the branched continued fraction
\[{\sum}_{i_{1=1}}^N\frac{a_{i(1)}}{1}+{\sum}_{i_{2=1}}^{i_1}\frac{a_{i(2)}}{1}+{\sum}_{i_{3=1}}^{i_2}\frac{a_{i(3)}}{1}+\cdots\]
whose elements belong to certain rectangular sets in the complex plane. The obtained results are applied to multidimensional \(S\)- and \(A\)-fractions with independent variables.A value region problem for continued fractions and discrete Dirac equations.https://www.zbmath.org/1456.300072021-04-16T16:22:00+00:00"Klimek, Slawomir"https://www.zbmath.org/authors/?q=ai:klimek.slawomir"McBride, Matt"https://www.zbmath.org/authors/?q=ai:mcbride.matt"Rathnayake, Sumedha"https://www.zbmath.org/authors/?q=ai:rathnayake.sumedha"Sakai, Kaoru"https://www.zbmath.org/authors/?q=ai:sakai.kaoruSummary: Motivated by applications in noncommutative geometry we prove several value range estimates for even convergents and tails, and odd reverse sequences of Stieltjes type continued fractions with bounded ratio of consecutive elements, and show how those estimates control growth of solutions of a system of discrete Dirac equations.Bohr-Rogosinski inequalities for bounded analytic functions.https://www.zbmath.org/1456.300012021-04-16T16:22:00+00:00"Alkhaleefah, Seraj A."https://www.zbmath.org/authors/?q=ai:alkhaleefah.seraj-a"Kayumov, Ilgiz R."https://www.zbmath.org/authors/?q=ai:kayumov.ilgiz-rifatovich"Ponnusamy, Saminathan"https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathanSummary: In this paper we first consider another version of the Rogosinski inequality for analytic functions \(f(z)=\sum_{n=0}^{\infty}a_nz^n\) in the unit disk \(|z|<1\), in which we replace the coefficients \(a_n\) (\(n=0,1,\ldots,N\)) of the power series by the derivatives \(f^{(n)}(z)/n!\) (\(n=0,1,\ldots,N\)). Secondly, we obtain improved versions of the classical Bohr inequality and Bohr's inequality for the harmonic mappings of the form \(f=h+\overline{g} \), where the analytic part \(h\) is bounded by 1 and that \(|g^{\prime}(z)|\leq k|h^{\prime}(z)|\) in \(|z|<1\) and for some \(k\in[0,1]\).Estimate for growth and decay of functions in Macintyre-Evgrafov kind theorems.https://www.zbmath.org/1456.300042021-04-16T16:22:00+00:00"Gaĭsin, Akhtyar Magazovich"https://www.zbmath.org/authors/?q=ai:gaisin.akhtyar-magazovich"Gaĭsina, Galiya Akhtyarovna"https://www.zbmath.org/authors/?q=ai:gaisina.galiya-akhtyarovnaSummary: In the paper we obtain two results on the behavior of Dirichlet series on the real axis.
The first of them concerns the lower bound for the sum of the Dirichlet series on the system of segments \([\alpha,\,\alpha+\delta]\). Here the parameters \(\alpha > 0\), \(\delta > 0\) are such that \(\alpha \uparrow + \infty\), \(\delta \downarrow 0\). The needed asymptotic estimates is established by means of a method based on some inequalities for extremal functions in the appropriate non-quasi-analytic Carleman class. This approach turns out to be more effective than the known traditional ways for obtaining similar estimates.
The second result specifies essentially the known theorem by M. A. Evgrafov on existence of a bounded on \(\mathbb{R}\) Dirichlet series. According to Macintyre, the sum of this series tends to zero on \(\mathbb{R} \). We prove a spectific estimate for the decay rate of the function in an Macintyre-Evgrafov type example.Schur's criterion for formal Newton series.https://www.zbmath.org/1456.300022021-04-16T16:22:00+00:00"Buslaev, V. I."https://www.zbmath.org/authors/?q=ai:buslaev.viktor-iFrom the introduction: \([\ldots]\) In the present note, we formulate an analog of Schur's criterion for the formal Newton series
\[
\Phi(z)=a_0+\sum_{k=1}^\infty a_k(z-e_1)\cdots(z-e_k)
\]in terms of determinants \([\ldots]\)