Approximation of Derivatives of Analytic Functions from One Hardy Class by Another Hardy Class

In the Hardy space Hp(Dϱ), 1 ≤ p ⪯ ∞, of functions analytic in the disk Dϱ = {z ∈ ℂ}: z < ϱ, we denote by NHp(Dϱ), N > 0, the class of functions whose Lp-norm on the circle γϱ = {z ∈ ℂ: z = ϱ} does not exceed the number N and by ∂Hp(Dϱ) the class consisting of the derivatives of functions from 1Hp(Dϱ). We consider the problem of the best approximation of the class ∂Hp(Dϱ) by the class NHp(DR)N > 0, with respect to the Lp-norm on the circle γr, 0 < r < ρ < R. The order of the best approximation as N → +∞ is found: (Formula presented.) In the case where the parameter N belongs to some sequence of intervals, the exact value of the best approximation and a linear method implementing it are obtained. A similar problem is considered for classes of functions analytic in annuli. © 2020, Pleiades Publishing, Ltd.This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University)

Аналог теоремы Адамара и связанные экстремальные задачи на классе аналитических функций

We study several related extremal problems for analytic functions in a finitely connected domain G with rectifiable Jordan boundary Γ. A sharp inequality is established between values of a function analytic in G and weighted means of its boundary values on two measurable subsets γ1 and γ0 = Γ \ γ1 of the boundary: |f(z0)| ≤ C kfkαLqϕ1 (γ1) kfkβLpϕ0 (γ0), z0 ∈ G, 0 < q, p ≤ ∞. The inequality is an analog of Hadamard’s three-circle theorem and the Nevanlinna brothers’ theorem on two constants. In the case of a doubly connected domain G and 1 ≤ q, p ≤ ∞, we study the cases where the inequality provides the value of the modulus of continuity for a functional of analytic extension of a function from a part of γ1 to a given point of the domain. In these cases, the corresponding problems of optimal recovery of a function from its approximate boundary values on γ1 and of the best approximation of a functional by linear bounded functionals are solved. The case of a simply connected domain G has been completely investigated previously. © 2020 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University), and as part of research conducted in the Ural Mathematical Center

Analog of the Hadamard Theorem and Related Extremal Problems on the Class of Analytic Functions

We study several related extremal problems for analytic functions in a finitely connected domain G with rectifiable Jordan boundary Γ. A sharp inequality is established between values of a function analytic in G and weighted means of its boundary values on two measurable subsets (Formula presented.) of the boundary: (Formula presented.) The inequality is an analog of Hadamard’s three-circle theorem and the Nevanlinna brothers’ two-constant theorem.In the case of a doubly connected domain G and (Formula presented.), we study the cases where the inequality provides the value of the modulus of continuity for a functional of analytic extension of a function from the part (Formula presented.) of the boundary to a given point of the domain. In these cases, the corresponding problem of optimal recovery of a function from its approximate boundary values on (Formula presented.) and the problem of the best approximation of a functional by bounded linear functionals are solved.The case of a simply connected domain G has been completely investigated previously. © 2021, Pleiades Publishing, Ltd.Russian Foundation for Basic Research, РФФИ: 18-01-00336This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University) and is a part of the research carried out at the Ural Mathematical Center

Approximation of derivatives of analytic functions from one Hardy class by another Hardy class

In the Hardy space Hp(De), 1 ≤ p ≤ ∞, of functions analytic in the disk De = {z ∈ C: |z| < e}, we denote by NHp(De), N > 0, the class of functions whose Lp-norm on the circle γe = {z ∈ C: |z| = e} does not exceed the number N and by ∂Hp(De) the class consisting of the derivatives of functions from 1Hp(De). We consider the problem of the best approximation of the class ∂Hp(Dρ) by the class NHp(DR), N > 0, with respect to the Lp-norm on the circle γr, 0 < r < ρ < R. The order of the best approximation as N → +∞ is found: (Equation presented) In the case where the parameter N belongs to some sequence of intervals, the exact value of the best approximation and a linear method implementing it are obtained. A similar problem is considered for classes of functions analytic in rings. © 2019 Mofid University - Center for Human Rights Studies. All rights reserved.Russian Foundation for Basic Research, RFBR: 18-01-00336Ministry of Education and Science of the Russian Federation, MinobrnaukaUral Federal University, UrFUThis work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University)

APPROXIMATION OF THE DIFFERENTIATION OPERATOR ON THE CLASS OF FUNCTIONS ANALYTIC IN AN ANNULUS

In the class of functions analytic in the annulus $C_r:=\left\{z\in\mathbb{C}\, :\, r<|z|<1\right\}$ with bounded $L^p$-norms on the unit circle, we study the problem of the best approximation of the operator taking the nontangential limit boundary values of a function on the circle $\Gamma_r$ of radius $r$ to values of the derivative of the function on the circle $\Gamma_\rho$ of radius $\rho,\, r<\rho<1,$ by bounded linear operators from $L^p(\Gamma_r)$ to $L^p(\Gamma_ \rho)$ with norms not exceeding a number $N$.  A solution of the problem has been obtained in the case when $N$ belongs to the union of a sequence of intervals. The related problem of optimal recovery of the derivative of a function from boundary values of the function on $\Gamma_\rho$ given with an error has been solved

Задача Стечкина о наилучшем приближении неограниченного оператора ограниченными и родственные ей задачи

This paper discusses Stechkin’s problem on the best approximation of a linear unbounded operator by bounded linear operators and related extremal problems. The main attention is paid to the approximation of differentiation operators in Lebesgue spaces on the axis and to the operator of the continuation of an analytic function to a domain from a part of the boundary of the domain. This is a review paper based on the materials of the authors’ lecture on September 14, 2020, at the X Internet video-conference “Day of Mathematics and Mechanics” of four institutes of the Russian Academy of Sciences: Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of RAS (Yekaterinburg), Sobolev Institute of the Siberian Branch of RAS (Novosibirsk), Steklov Mathematical Institute (Moscow), and the St. Petersburg Department of the Steklov Mathematical Institute. The lecture of the authors was dedicated to the 100th anniversary of the birth of Sergei Borisovich Stechkin. The problem of the best approximation of a linear unbounded operator by bounded ones is one of his legacies. We tried to at least partially reflect the new results, methods, and statements that appeared in this topic after the publication of the review papers (Arestov, Gabushin, 1995–1996). The material on this topic is wide; the selection of the material for the lecture and paper is the responsibility of the authors. © 2020 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.This work was performed as a part of the research conducted in the Ural Mathematical Center and also supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University)

Using of turquoise organization’s concept to benchmark indicators of creative team management

The purpose of this study is to study the possibilities of applying the turquoise type organization model in creative industries, where there is a need for constant staff development to achieve maximum results. When analyzing the turquoise organization model, a number of factors should be taken into account, on which it relies, and also choose from a number of them those that really have a significant impact on the formation of the turquoise approach in creative industries. At a certain stage, from a non-standard creative management model, one way or another, it is necessary to return to traditional management methods while maintaining the basic principles of the turquoise organization, which can be considered as a standard. The introduction of such tools into the creative team/process management system qualitatively improves the mechanism of managerial decision-making, and also allows it to be debugged through unidirectional actions of both managers and the entire creative team. To implement the task of standardization, a universal model of quantitative assessment of qualitative indicators was developed, taking into account the dynamics of their changes. At the same time, a model was built for selecting indicators for the formation of a set of personnel characteristics in order to benchmark them relative to the level of the same characteristics of the turquoise organization, taking into account the possibility of their transversal change over time

Ontologization of mental representations for management purposes

The article considers the problems of ontologization of mental representations of a certain field of activity and the corresponding subject area for the purposes of management automation. For the first time, the authors raise the question about the need to include mental representations of the sphere of activity that the subject area reflects in the ontology of the subject area. Ontologization of mental representations is necessary to identify the participants' understanding of their actions, which will change approaches to management, and for deep training of artificial intelligence in order to transfer part of its management functions to it. On the basis of connectionist approaches to understanding thinking, the paper considers the processes of forming mental representations and changing their attributes. The article gives a new definition and typology of mental representations

STUDY OF THE INFLUENCE OF FACTORS ON THE CONTENT OF RUSSIAN FEDERAL CHANNELS IN THE CONTEXT OF DIGITALIZATION

The authors carry out a study of the influence of various factors on the content of Russian federal channels. Using content analysis of the broadcast data were obtained to identify the influence of factors, characteristic for the stage of formation of a consumer society in Russia on the content of television channels and the analysis of conformity of telecasts the authors developed, criteria that have been formed and considered together with current trends in society related to the digitalization processes, and preferences of Russians. Television is an important social institution that should not only reflect the current situation in society, but also broadcast certain behaviors, culture and values. Television content requires a lot of attention, because now there are significant changes in the individual in the new conditions of digital communication. Modern television content should move away from the simple model of mass culture, which is based on a standardized approach based on entertainment content. Content should take into account modern consumer values