On Nikol'skii Type Inequality between the Uniform Norm and the Integral q-norm with Laguerre Weight of Algebraic Polynomials on the Half-Line

We study the Nikol'skii type inequality for algebraic polynomials on the half-line [0,∞) between the “uniform” norm sup{|f(x)|e−x∕2:x∈[0,∞)} and the norm ∫0 ∞|f(x)e−x∕2|qxαdx1∕q of the space Lα q with the Laguerre weight for 1≤q<∞ and α≥0. It is proved that the polynomial with a fixed leading coefficient that deviates least from zero in the space Lα+1 q is the unique extremal polynomial in the Nikol'skii inequality. To prove this result, we use the Laguerre translation. The properties of the norm of the Laguerre translation in the spaces Lα q are studied. © 2017 Elsevier Inc.This work was supported by the Russian Foundation for Basic Research (Project No. 15-01-02705), by the Program for State Support of Leading Scientific Schools of the Russian Federation (Project no. NSh-9356.2016.1), and by the Competitiveness Enhancement Program of the Ural Federal University (Enactment of the Government of the Russian Federation of March 16, 2013 No. 211, Agreement No. 02.A03.21.0006 of August 27, 2013)

One-sided weighted integral approximation of characteristic functions of intervals by polynomials on a closed interval

We consider the problem of one-sided weighted integral approximation on the interval [−1, 1] to the characteristic functions of intervals (a, 1] ⊂ (−1, 1] and (a, b) ⊂ (−1, 1) by algebraic polynomials. In the case of half-intervals, the problem is solved completely. We construct an example to illustrate the difficulties arising in the case of an open interval. © 2017, Pleiades Publishing, Ltd

O международной школе-конференции по теории функций, посвященной 100-летию со дня рождения С. Б. Стечкина

The paper provides an overview of the main events of the International Workshop–Conference on Function Theory dedicated to the Centenary of the birth of S. B. Stechkin, which was held in Yekaterinburg online from August 3 to August 12, 2020, for the 45th time since 1971. A description of the traditions and peculiarities of such workshops that have developed over the years as well as a list of reports by the conference participants are given. The paper also contains memoirs about Sergei Borisovich Stechkin, the initiator of such workshops–conferences, the founder and head of the scientific school on function theory. © 2020 Krasovskii Institute of Mathematics and Mechanics. All rights reserved

Several extremal approximation problems for the characteristic function of a spherical layer

We discuss three interrelated extremal problems on the set P n,m of algebraic polynomials of a given degree n on the unit sphere S m-1 of the Euclidean space ℝ m of dimension m ≥ 2. (1) Find the norm of the functional F(η) = F hP n = ∫ G(η)P n(x)dx, which is the integral over the spherical layer G(η) = {x = (x 1,...,x m) ∈ S m-1: h′ ≤ x m ≤ h″} defined by a pair of real numbers η = (h′, h″), -1 ≤ h′ < h″ ≤ 1, on the set P n,m with the norm of the space L(S m-1) of functions summable on the sphere. (2) Find the best approximation in L ∞(S m-1) of the characteristic function χ η of the layer G(η) by the subspace P n,m ⊥ of functions from L ∞(S m-1) that are orthogonal to the space of polynomials P n,m. (3) Find the best approximation in the space L(S m-1) of the function χ η by the space of polynomials P n,m. We present a solution of all three problems for the values h′ and h″ that are neighboring roots of the polynomial in a single variable of degree n + 1 that deviates least from zero in the space L 1 φ{symbol} (-1, 1) of functions summable on the interval (-1, 1) with ultraspherical weight φ{symbol}(t) = (1 - t 2) α, α = (m - 3)/2. We study the respective one-dimensional problems in the space of functions summable on (-1, 1) with an arbitrary not necessarily ultraspherical weight. © 2012 Pleiades Publishing, Ltd

Международная школа-конференция С.Б. Стечкина по теории функций, посвященная 85-летию чл.-корр. РАН Ю. Н. Субботина и заслуженного деятеля науки РФ Н.И. Черных

Brief information about S. B. Stechkin's workshopconference,on function theory for the fifty years of its existence,is given. The 46th workshop-conference dedicated to the 85th,anniversary of Corresponding Member of the Russian Academy,of Sciences Yu. N. Subbotin and Honored Scientist of the Russian,Federation N. I. Chernykh, which took place with face-to-face and virtual,participation in the Chemal region of the Altai Republic from August,9 to 19, 2021, is presented in more detail. The list of reports with a,summary of the participants of the workshop-conference is given © 2021, Siberian Electronic Mathematical Reports. All Rights Reserved.Работа по организации и проведению школы-конференции выполнена в рамках исследовательской и образовательной деятельности Уральского математического центра при финансовой поддержке Министерства науки и высшего образования РФ (номер соглашения 075-02-2021-1383)