Recent zbMATH articles in MSC 30Bhttps://www.zbmath.org/atom/cc/30B2022-05-16T20:40:13.078697ZWerkzeugBohr-type inequalities with one parameter for bounded analytic functions of Schwarz functionshttps://www.zbmath.org/1483.300042022-05-16T20:40:13.078697Z"Hu, Xiaojun"https://www.zbmath.org/authors/?q=ai:hu.xiaojun"Wang, Qihan"https://www.zbmath.org/authors/?q=ai:wang.qihan"Long, Boyong"https://www.zbmath.org/authors/?q=ai:long.boyongSummary: In this article, some Bohr-type inequalities with one parameter or involving convex combination for bounded analytic functions of Schwarz functions are established. Some previous inequalities are generalized. All the results are sharp.The Bohr inequality for the generalized Cesáro averaging operatorshttps://www.zbmath.org/1483.300052022-05-16T20:40:13.078697Z"Kayumov, Ilgiz R."https://www.zbmath.org/authors/?q=ai:kayumov.ilgiz-rifatovich"Khammatova, Diana M."https://www.zbmath.org/authors/?q=ai:khammatova.diana-m"Ponnusamy, Saminathan"https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathanSummary: The main aim of this paper is to prove a generalization of the classical Bohr theorem and as an application, we obtain a counterpart of Bohr theorem for the generalized Cesáro operator.Some Bohr-type inequalities with one parameter for bounded analytic functionshttps://www.zbmath.org/1483.300062022-05-16T20:40:13.078697Z"Wu, Le"https://www.zbmath.org/authors/?q=ai:wu.le"Wang, Qihan"https://www.zbmath.org/authors/?q=ai:wang.qihan"Long, Boyong"https://www.zbmath.org/authors/?q=ai:long.boyongSummary: In this paper, for bounded analytic function, some Bohr-type inequalities with one parameter or involving convex combination are established. Most of the results are sharp. Some previous results are generalized.A bilogarithmic criterion for the existence of a regular minorant that does not satisfy the bang conditionhttps://www.zbmath.org/1483.300072022-05-16T20:40:13.078697Z"Gaisin, R. A."https://www.zbmath.org/authors/?q=ai:gaisin.rashit-akhtyarovichSummary: Problems of constructing regular majorants for sequences \(\mu=\{\mu_n\}_{n=0}^{\infty}\) of numbers \(\mu_n\ge0\) that are the Taylor coefficients of integer transcendental functions of minimal exponential type are investigated. A new criterion for the existence of regular minorants of associated sequences of the extended half-line \((0,+\infty]\) in terms of the Levinson bilogarithmic condition \(M=\{\mu_n^{-1}\}_{n=0}^{\infty}\) is obtained. The result provides a necessary and sufficient condition for the nontriviality of the important subclass defined by J. A. Siddiqi. The proofs of the main statements are based on properties of the Legendre transform.Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applicationshttps://www.zbmath.org/1483.300082022-05-16T20:40:13.078697Z"Do, Yen Q."https://www.zbmath.org/authors/?q=ai:do.yen-qSummary: This paper seeks to further explore the distribution of the real roots of random polynomials with non-centered coefficients. We focus on polynomials where the typical values of the coefficients have power growth and count the average number of real zeros. Almost all previous results require coefficients with zero mean, and it is non-trivial to extend these results to the general case. Our approach is based on a novel comparison principle that reduces the general situation to the mean-zero setting. As applications, we obtain new results for the Kac polynomials, hyperbolic random polynomials, their derivatives, and generalizations of these polynomials. The proof features new logarithmic integrability estimates for random polynomials (both local and global) and fairly sharp estimates for the local number of real zeros.Riesz projection and bounded mean oscillation for Dirichlet serieshttps://www.zbmath.org/1483.300092022-05-16T20:40:13.078697Z"Konyagin, Sergei"https://www.zbmath.org/authors/?q=ai:konyagin.sergey-v"Queffélec, Hervé"https://www.zbmath.org/authors/?q=ai:queffelec.herve"Saksman, Eero"https://www.zbmath.org/authors/?q=ai:saksman.eero"Seip, Kristian"https://www.zbmath.org/authors/?q=ai:seip.kristianSummary: We prove that the norm of the Riesz projection from \(L^\infty (\mathbb{T}^n)\) to \(L^p(\mathbb{T}^n)\) is \(1\) for all \(n\ge 1\) only if \(p\le 2\), thus solving a problem posed by \textit{J. Marzo} and the fourth author [Bull. Sci. Math. 135, No. 3, 324--331 (2011; Zbl 1221.42013)]. This shows that \(H^p(\mathbb{T}^{\infty})\) does not contain the dual space of \(H^1(\mathbb{T}^{\infty})\) for any \(p > 2\). We then note that the dual of \(H^1(\mathbb{T}^{\infty})\) contains, via the Bohr lift, the space of Dirichlet series in BMOA of the right half-plane. We give several conditions showing how this BMOA space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on \(\mathbb{T}\), we compute its \(L^p\) norm when \(1 < p < \infty\), and we use this result to show that the \(L^\infty\) norm of the \(N\) th partial sum of a bounded Dirichlet series over \(d\)-smooth numbers is of order \(\log\log N\).On conditions of the completeness of some systems of Bessel functions in the space \(L^2 ((0;1); x^{2p} dx)\)https://www.zbmath.org/1483.420232022-05-16T20:40:13.078697Z"Khats, R. V."https://www.zbmath.org/authors/?q=ai:khats.r-vIn this paper the author gives necessary and sufficient conditions for the system \(\{x^{-p-1}\sqrt{x\rho_k}J_\nu(x\rho_k): k \in \mathbb{N}\}\) to be complete in the weighted space \(L^2((0,1), x^{2p} dx)\). Here \(J_\nu\) is the first kind Bessel function of index \(\nu \geq \frac{1}{2}\), \(p \in \mathbb{R}\) and \(\rho_k : k \in \mathbb{N}\) is an arbitrary sequence of distinct nonzero complex numbers.
The fact that \(\rho_k\) can be arbitrary had already been considered by \textit{B. V. Vynnyts'kyi} and \textit{R. V. Khats'} [Eurasian Math. J. 6, No. 1, 123--131 (2015; Zbl 1463.30015)]. In the present paper, he gives new conditions which depend only on properties of the \(\rho_k\).
Reviewer: Ursula Molter (Buenos Aires)Riesz means in Hardy spaces on Dirichlet groupshttps://www.zbmath.org/1483.430062022-05-16T20:40:13.078697Z"Defant, Andreas"https://www.zbmath.org/authors/?q=ai:defant.andreas"Schoolmann, Ingo"https://www.zbmath.org/authors/?q=ai:schoolmann.ingoIn a recent series of papers the authors have developed a theory of Hardy spaces of general Dirichlet series, closely connected with harmonic analysis on groups. Given a frequency \(\lambda = (\lambda_{n})_{n}\) (i.e., strictly increasing and unbounded), they introduced in [\textit{A. Defant} and \textit{I. Schoolmann}, J. Fourier Anal. Appl. 25, No. 6, 3220--3258 (2019; Zbl 1429.43004)] the notion of \(\lambda\)-Dirichlet group (which defines a family of characters \((h_{\lambda_{n}})_{n}\)). For such a group \(G\), they also defined the Hardy space \(H_{p}^{\lambda}(G)\) for \(1 \leq p \leq \infty\). Here they deal with the convergence of the Riesz means for functions in these spaces.
Given \(f \in H_{1}^{\lambda}(G)\), the first \((\lambda,k)\)-Riesz sum of length \(x >0\) is defined as
\[
R^{\lambda,k}_{x}(f) = \sum_{\lambda_{n}<x} \hat{f}(h_{\lambda_{n}}) \Big( 1 - \frac{\lambda_{n}}{x} \Big)^{k} h_{\lambda_{n}} \,.
\]
The main result of the paper shows that, for every \(k>0\), the expression
\[
R^{\lambda,k}_{\max}(f) (\omega) = \sup_{x >0} \big\vert R^{\lambda,k}_{x}(f) (\omega) \big\vert \,,
\]
for \(f \in H_{1}^{\lambda}(G)\) and \(\omega \in G\), defines a bounded sublinear operator
\[
R^{\lambda,k}_{\max} : H_{1}^{\lambda}(G) \to L_{1,\infty}(G)
\]
and
\[
R^{\lambda,k}_{\max} : H_{p}^{\lambda}(G) \to L_{p}(G) \text{ for } 1 < p \leq \infty .
\]
As a consequence, \(R^{\lambda,k}_{x}(f)(\omega)\) converges (in \(x\)) to \(f(\omega)\) for almost every \(\omega\).
When horizontal translations are considered, the situation improves. It is shown that for \(u,k > 0\), there exists a constant \(C=C(u,k)\) so that for every frequency \(\lambda\), all \(1 \leq p \leq \infty\) and \(f \in H_{p}(G)^{\lambda}\) we have
\[
\bigg( \int_{G} \sup_{x >0} \Big\vert \sum_{\lambda_{n} < x} \hat{f} (h_{\lambda_{n}}) e^{-u\lambda_{n}} \Big( 1 - \frac{\lambda_{n}}{x} \Big)^{k} h_{\lambda_{n}} (\omega) \Big\vert^{p} d \omega \bigg)^{1/p} \leq C \Vert f \Vert_{p} \,.
\]
Note that in this case the inequality holds even for \(p=1\), and that the constant does not depend on \(p\).
One of the main tools to prove the main result is a maximal Hardy-Littlewood operator, adapted to this setting. If \((G, \beta)\) is a Dirichlet group and \(f \in L_{1}(G)\), then for almost every \(\omega \in G\) the function defined by \(f_{\omega}(t) = f(\omega \beta(t))\) is locally integrable on \(\mathbb{R}\). It is proved that the adapted Hardy-Littlewood maximal operator, given by
\[
\overline{M}(f) (\omega) = \sup_{\genfrac{}{}{0pt}{2}{I \subset \mathbb{R}}{\text{interval}}} \frac{1}{\vert I \vert} \int_{I} \vert f_{\omega} (t) \vert dt
\]
defines a sublinear bounded operator \(\overline{M}: L_{1}(G) \to L_{1,\infty}(G)\) and \(\overline{M}: L_{p}(G) \to L_{p}(G)\) for \(1 < p \leq \infty\).
It is known that, for \(1 < p < \infty\) and any frequency \(\lambda\), the sequence \((h_{\lambda_{n}})\) is a Schauder basis of \(H_{p}^{\lambda}(G)\) and, therefore the Riesz means of any function \(f\) converge (in norm) to \(f\). Here it is proved that this is also the case for \(p=1\), that is
\[
\lim_{x \to \infty} \big\Vert R^{\lambda, k}_{x}(f) - f \Vert_{1} =0
\]
for every \(k>0\) and every \(f \in H_{1}^{\lambda}(G)\).
Applications of all these are given to general Dirichlet series and to almost periodic functions.
Reviewer: Pablo Sevilla Peris (Valencia)Invariant spaces of entire functionshttps://www.zbmath.org/1483.460212022-05-16T20:40:13.078697Z"Krivosheev, A. S."https://www.zbmath.org/authors/?q=ai:krivosheev.aleksandr-sergeevich"Krivosheeva, O. A."https://www.zbmath.org/authors/?q=ai:krivosheeva.o-aLet \(D\subset\mathbb{C}\) be a convex domain and let \(H(D)\) be the space of holomorphic functions on \(D\) endowed with the compact open topology. The paper under review deals with the following problem: Let \(W\) be an invariant subspace of the differentiation operator on \(H(D)\). Which conditions ensure that all functions of \(W\) can be extended to entire functions? This problem naturally arises from the problem of expanding convergence domains of exponential series and their special cases, power series and Dirichlet series. \(W\) is assumed to satisfy \textit{spectral synthesis}, i.e., the closure of the span of the eigenvectors of the differentiation operator in \(H(D)\) is the whole \(W\). The following subset of the unit circle \(\mathbb{T}\) is defined,
\[
J(D)= \Bigl\{\omega\in \mathbb T: \ \sup_{z\in D}\text{Re}\,z\omega=+\infty \Bigr\}.
\]
Let \(\Delta:=\{\lambda_k: k\in\mathbb{N}\}\) be the sequence of eigenvalues of the differentiation operator acting on \(W\). Let \(\Xi(\Delta):=\{\overline{\lambda}/|\lambda|: \lambda\in \Delta\} \). The main theorem asserts that the continuation problem has a positive solution when \(\Xi(\Delta)\subset J(D)\). This result was known only under the assumption that \(J(D)\) is open in \(\mathbb{T}\).
Reviewer: Enrique Jordá (Alicante)Norms of composition operators on the \(H^2\) space of Dirichlet serieshttps://www.zbmath.org/1483.470452022-05-16T20:40:13.078697Z"Brevig, Ole Fredrik"https://www.zbmath.org/authors/?q=ai:brevig.ole-fredrik"Perfekt, Karl-Mikael"https://www.zbmath.org/authors/?q=ai:perfekt.karl-mikaelSummary: We consider composition operators \(\mathscr{C}_\varphi\) on the Hardy space of Dirichlet series \(\mathscr{H}^2\), generated by Dirichlet series symbols \(\varphi \). We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of \(\varphi \). The other concerns general symbols, and is based on a geometrical condition on the boundary values of \(\varphi \). Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particular, we generalize a result of \textit{J. H. Shapiro} [Monatsh. Math. 130, No. 1, 57--70 (2000; Zbl 0951.47026)] on the norm of composition operators on the classical Hardy space of the unit disc. Based on our techniques, we also improve the recently established upper and lower norm bounds in the special case that \(\varphi(s) = c + r 2^{- s} \). A~number of other examples are given.