A sum-product estimate in fields of prime order

Let q be a prime, A be a subset of a finite field $F=\Bbb Z/q\Bbb Z$, $|A|<\sqrt{|F|}$. We prove the estimate $\max(|A+A|,|A\cdot A|)\ge c|A|^{1+\epsilon}$ for some $\epsilon>0$ and c>0. This extends the result of J. Bourgain, N. Katz, and T. Tao

Delta-semidefinite and delta-convex quadratic forms in Banach spaces

A continuous quadratic form ("quadratic form", in short) on a Banach space $X$ is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator $T\colon X\to X^*$ factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if $T$ is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional $L_p(\mu)$ space ($1\le p \le\infty$) is: (a) delta-semidefinite iff $p \ge 2$; (b) delta-convex iff $p>1$. Some other related results concerning delta-convexity are proved and some open problems are stated.Comment: 19 page

Greedy Approximation with Regard to Bases and General Minimal Systems

*This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003This paper is a survey which also contains some new results on the nonlinear approximation with regard to a basis or, more generally, with regard to a minimal system. Approximation takes place in a Banach or in a quasi-Banach space. The last decade was very successful in studying nonlinear approximation. This was motivated by numerous applications. Nonlinear approximation is important in applications because of its increased efficiency. Two types of nonlinear approximation are employed frequently in applications. Adaptive methods are used in PDE solvers. The m-term approximation considered here is used in image and signal processing as well as the design of neural networks. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a fixed linear space but are allowed to depend on the function being approximated. The fundamental question of nonlinear approximation is how to construct good methods (algorithms) of nonlinear approximation. In this paper we discuss greedy type and thresholding type algorithms

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

Continuous selections of multivalued mappings

This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II, which was published in 2002. In comparison, our present survey considers more restricted and specific areas of mathematics. Note that we do not consider the theory of selectors (i.e. continuous choices of elements from subsets of topological spaces) since this topics is covered by another survey in this volume

Small doubling in groups

Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous Freiman--Ruzsa theorem.Comment: 23 pages, survey article submitted to Proceedings of the Erdos Centenary conferenc