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COMMON PART

Project Number21-11-00131

Project titleExtremal problems in the theory of orthogonal series, approximation theory and complex analysis

Project LeadKashin Boris

AffiliationFederal State Budgetary Educational Institution of Higher Education Lomonosov Moscow State University,

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-109 - Substantial and functional analysis

Keywordswidth, submatrix, operator norm, discretization, orthonormal system, trigonometric series, Weyl sums, Fourier transform, generalized monotony, modulus of continuity, holomorphic mappings, fixed points, domains of univalence, greedy approximations

PROJECT CONTENT

Annotation
The first sets of extremal problems were related to the period of establishment of Mathematics as a science, and they were essentially connected to practical human activities. Nevertheless, over time the questions of extremal values of various quantitative features or of their quite precise estimates have appeared in the majority of the areas of Mathematics. Moreover, they often refer to more natural subjects, satisfying common standards of beauty and significance. The present project is devoted to solving some extremal problems of function theory and functional analysis. Besides, it is supposed to pay considerable attention to possible applications of the obtained results. It is worth mentioning that the importance of applications of various areas of function theory and functional analysis has significantly increased during last 10-15 years. The results which used to be seen as purely theoretical happened to be needed in practice. The reason was the fast development of information technology and the necessity to create optimal algorithms, which would be able to handle large amount of information. A clear illustration directly connected to the present project is so-called «compressed sensing», which is a very quickly developing direction in working with data. The results of approximation theory about estimates of the Kolmogorov width, obtained in Soviet Union back in 70-80 of the previous century, became a mathematical fundament for this area. Nowadays there are tens of thousands of publications on this topic, and the algorithms using «compressed sensing» are widely brought into practice (unfortunately, not in Russia). One of the important components for some of these algorithms — the so-called «greedy approximations» is also planned to be investigated within the project. There is another topic, which is important for both theoretical questions and applications, namely, investigation of submatrices with determined extremal properties of a given matrix of a high dimension. This direction appeared also in the Soviet Union, in view of not practical problems of almost everywhere convergence of orthogonal series, and developed into a self-contained area of functional analysis. Its results turned out to be sought after in information technology for solving problems of a graph sparsification (i.e., of substitution of a big graph by its small subgraph with similar spectral properties). They also allow providing acceleration of classical algorithms of solving systems of linear equations in high dimension. The participants of the project have lately obtained a number of important results on submatrices with extremal properties. One of the problems of the project is to investigate this topic extensively. Studying the properties of submatrices of a given matrix is directly connected to one more topic which is to be studied during the project - that is discretization on functional systems. Classical results of theory of trigonometric series states equivalence of the norm of a trigonometric polynomial of degree N (in the most important spaces for theory and practice) and the corresponding discrete analogue, calculated using the values of the polynomial on the uniform grid with the number of lattice points of the order N. In the case of considering some other important finite-dimensional spaces of function instead of the space of trigonometric polynomials, the problem of finding a grid with the minimal possible number of elements using the values of a function, on which it would be possible to strictly estimate its original norm, appears in various theoretical and applied questions, but it turns out to be much more complicated. The next range of extremal problems which are planned to be studied during the project is related to the classical perspective of convergence of one-dimensional and multiple trigonometric series. Here we have strict estimates, using analytical methods, of one-dimensional and multiple trigonometric polynomials, based on their spectra and coefficients. In the important for applications case of one-dimensional polynomials with a polynomial spectrum the participants of the project have obtained the results that are close to final ones. One more essential direction which is planned to be studied is related to the problems of finding domains of univalence of analytic functions. Conditions of univalence and domains of univalence of analytic functions belong to traditionally important topics of geometrical function theory. Univalence implies a number of other geometrical and analytic properties of a function. It is often connected to physical realization of a mathematical model in applications. That is why the topic is so relevant and significant in both theory and applications. Such invariants as fixed points of a mapping play an important role in research of this area. The first results on the sharp domain of univalence on the class of functions mapping the unit disc into itself and having an interior fixed point were obtained in the 20s of the previous century. Further, it turned out that a holomorphic mapping of the unit disc into itself with two fixed points (one of them in nontrivial case should lie on the boundary) has domains of univalence for some values of the angular derivative at the fixed point on the boundary. The participants of the project have recently managed to extend domain of univalence on the class of functions with an interior and a boundary fixed points and investigate the question of preciseness of the obtained domains of univalence. Success in solving this problem is associated with the effective penetration of real analysis methods into the problems of the theory of functions of a complex variable discovered by the project participants. During the project it is planned to obtain the final solution to the extremal problem of finding the domain of univalence on a given class of functions. Thus, modern research of domains of univalence will be finalized and some classical results will be clarified. The methods planned to involve proceeding with problems, which are supposed to be studied during this project, are quite diverse. In addition to the classical tools of function theory it is necessary to apply modern methods of high-dimensional geometry, probability theory, combinatorics, number theory, complex analysis. Besides, the use of known results is not sufficient for achieving the goals of the project. The participants of the project are supposed to get advances in the mentioned areas.

Expected results
We intend to publish the main publications in the central Russian mathematical journals “Russian Mathenatical Surveys ”, “Izvestiya: Mathematics”, “Sbornik: Mathematics”, “Mathematical Notes”, “Proceedings of the Steklov Institute of Mathematics”, “St. Petersburg Mathematical Journal”. For some tasks, international cooperation is expected, in these cases publications will be presented in high-rating foreign journals. The authors of the project plan to make at least 10 reports at international conferences held both in Russia and abroad annually. It is planned to obtain the following results during the project. We will study the approximate properties when approximating essentially nonlinear functional classes and nonlinear operators acting between Banach spaces X and Y by linear subspaces. In the latter case, the answer is given in terms of the table of Kolmogorov widths that characterize the error in approximating the image of a unit ball of an arbitrary n-dimensional subspace of X by an N-dimensional subspace of Y. Problems of this type are important both for function theory and for computer science and are actively studied by mathematicians in the USA.,China and Europe, since they find applications when working with "big data". Project Manager results on estimates of widths are fundamental and expected results undoubtedly will be world-class. It is planned to publish 3 articles on this topic. For a given N x n matrix A, n <N, considered as an operator from the space l_p ^ n to l_q ^ N, 1 \ leqslant p, q \ leqslant \ infty, we plan to obtain estimates for the norms of m x n submatrices that depend on the parameter m. This problem is important for applications in various issues of functional analysis, computational mathematics and theoretical computer science. But exact estimates are known only for the case p = q = 2. For other values of the parameters p and q, the main known results were obtained by project participants B.S. Kashin and I.V. Limonova. The goal of the project is to bring these results to the final and give their application to the issues of discretization of functional systems. On the topic is planned 3 publications. Greedy approximations is an approach that is widely used in applications. However, there systematic study at the theoretical level began, in fact, only at the end of XX century. It is expected to obtain sharp conditions on a sequence of errors in calculating coefficients (in the case of approximate greedy expansions) and on a sequence of coefficients (in the case of greedy expansions with prescribed coefficients), which guarantee the convergence of the corresponding greedy expansion to an expanded element. This will eliminate the "gray" zone between both positive and negative convergence results, which still exists for these types of greedy expansions. In addition to the theoretical interest, these results are also of applied interest since approximate greedy expansions are one of the most used types of greedy approximations from an applied point of view. It is planned to publish 2 articles on this topic. The well-known theorem of D.E. Menshov states that it is possible to construct an orthonormal system for which any monotone sequence a_n = o (\ log_2 ^ 2 n) is not a Weyl multiplier. D.E. Menshov and a number of other authors used the properties of the Hilbert matrix to construct such orthonormal systems. We plan to investigate the problem of finding orthonormal systems satisfying Menshov's theorem, the construction of which has a fundamentally different nature and does not use the properties of the Hilbert matrix. A discrete orthonormal system with an extremely large L_2-norm of the maximal operator will be found. In this case, the matrix defining the orthonormal system will have improved characteristics in comparison with the Hilbert matrix. At least 1 publication is planned on this topic. It will be studied in the field of trigonometric series asymptotic behavior near the origin of the sum of sine series of the form g(b,x)=\sum (k=1)^\infty b k\sin kx with a convex slowly varying sequence b. The case of slowly varying coefficients is the most fundamental. Then the sequence of partial sums of the sine series converges very slowly, and its sum g(b,x) could turn out to be non-integrable. The main term of asymptotic behavior of the sum of sine series with convex slowly varying coefficients was found by Aljancic, Bojanic and Tomic in 1956. For estimating the difference of g(b,x) from the main term of its asymptotic behavior b_{m(x)}/x, m(x)=[\pi/x], S.A. Telyakovsky used the piecewise continuous function x\sum_{k=1}^{m(x)-1}k^2(b_k-b_{k+1}). He showed that in some neighborhood of zero the difference g(b,x)-b_{m(x)}/x can be estimated from both sides by the mentioned function with an absolute constant, independent of b. In 2020 the participant of the project A.P. Solodov presented the sharp values of that constants. During the project asymptotic of the function g(b,x) on the class of convex slowly varying sequences will be found in the regular case. It is planned to publish at least 1 article on this topic. Another direction of researches in theory of trigonometric series is to introduce a new class of integrable functions on a line, which would be a contuinuous analogue of the class of trigonometric series with general monotone coefficients. It is well known that trigonometric series with monotone coefficients possess a number of valuable properties (for instance, criteria of belonging of such series to classes L р and H^(\omega)). However, this class is rather narrow, which motivated for publications of works on extending this class. The most successful way of extending happened to be the so-called GM class of cosine or sine series, whose coefficients for some C>1 and some natural \nu>1 for any natural N\geq \nu satisfy the inequality \sum\limits_{n=2^ N}^{2^(N+1)} |a_n —а_{n+1}| \leq C2^{-N}\sum\limits_{n=2^{N-\nu}}^{2^{N+\nu}} |a_n|. Nevertheless, till last days authors had to require nonnegativity of the coefficients as well. This limitation was removed due to the joint works of M.I. Dyachenko, S.Yu. Tikhonov, A.B. Mukanov published in 2017-2020. Now it is supposed to be an actual problem to introduce an analogous class and obtain corresponding results in the non-periodic case. It is planned to publish 3 papers on this topic. It is planned to obtain a description of the set of zeros on the circle of a power series with an absolutely convergent series of coefficients. It is also planned to improve the John-Nirenberg inequality for Riemann type series \sum\limits_{n=1}^{\infty}e^{2\pi i n^k x}/n of degree k>2, based on previously obtained estimates of Weyl sums at points poorly approximated by rational numbers. The corresponding result for the Riemann series of degree two was presented in 2018 (Chamizo, Córdoba, Ubis). Riemann type series are found in many problems of harmonic analysis, and it is very important to understand how far these functions are from being bounded. It is planned to publish 2 papers on this topic. Studies of the problem of sharp domains of univalence have been carried out since the 1920s, and the first results are due to E. Landau and J. Valiron. Fixed points play an important role here. Landau's result works for about a class of functions with one fixed point inside the unit disc. In the future, similar results on domains of univalence were generalized to classes of functions with two fixed points (at least one of which lies on the boundary). During the project, sharp domains of univalence will be obtained on the class of holomorphic mappings of the disc into itself with interior and boundary fixed points that have a restriction on the value of the angular derivative at the boundary fixed point. These results will be an important addition to the classical theorem of E. Landau. On the class of holomorphic maps of the unit disc into itself with a boundary fixed point, the well-known Julia – Carathéodory inequality gives a lower estimate for the angular derivative at the boundary fixed point. In the course of the project, a strengthening of the Julia – Carathéodory inequality will be obtained on the subclass of holomorphic functions that map the unit disc into itself, the point z = a to the point z = b, when there is information about the value of the derivative at the point z = a. We will solve the problem of describing the sharp set of values of the nth-order derivative at an internal fixed point on the class of holomorphic functions that map the unit disc into itself, with an internal and an arbitrary finite set of boundary fixed points. This result will complement the classical Schwarz lemma, which plays a fundamental role in complex analysis and has numerous applications in geometric function theory. It is planned to publish 4 papers on this topic. The problem of density of an additive semigroup in a Banach space X will be investigated. This task was first posed by P. A. Borodin in 2014. It consists in finding necessary and sufficient conditions on a set M in a Banach space X, which are sufficient for the additive semigroup R (M) generated by M to be dense in the space X. Expected results will be used for the approximation by different systems of functions in different spaces. In particular, the problem of approximating harmonic functions by functions of the form | x-a | ^ {- n + 2} + const with poles on surfaces in R ^ n will be solved. This problem is a natural multidimensional analogue of the approximation by the simplest fractions and has been solved so far only in the case of a smooth surface in R ^ 3 in the work of D. T. Piele. At least 1 publication is planned on this topic. The planned results correspond the international level of achievements in function theory and functional analysis. A lot of results will become final: during the research of any quantity either its precise value will be given or two-sided estimates (each one of which will be unimprovable) will be provided.

REPORTS