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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

Keywordsm-term approximation, Banach space, dictionary, greedy algorithm, semigroup, simplest fraction, metric projection, Chebyshev set

Annotation
The aim of the project is to develop nonlinear approximation theory and geometry of Banach spaces in their natural relationship. More detailed: investigation of the properties of m-term approximations depending on the geometry of the dictionary; ascertainment of dichotomy for sequences of the least m-term deviations; proof that not every strictly monotone zero-tending sequence can be a sequence of rational deviations in the Euclidean norm; development of the theory of density of semigroups in Banach spaces; applications to the approximation by simple partial fractions in the complex domain; investigation of the convergence of greedy algorithms depending on the geometry of the dictionary and the enclosing Banach space; improvement of the known estimates for the convergence rate of greedy approximation with respect to special dictionaries; introduction and study of the convergence of greedy approximations with respect to several dictionaries; investigation of the problem of weak convergence of consecutive random projections on convex sets; progress in the classical directions of geometric approximation theory, in particular, in the problem of monotone linear connectivity of Chebyshev sets in a finite-dimensional normed space. All these problems are relevant and attract the most serious attention of specialists in modern functional analysis, approximation theory and geometry. All the expected results will be new and should make a significant contribution to the approximation theory.

Expected results
Here are some of the expected results: establishing a dichotomy for sequences of the least m-term deviations; proving that not every strictly monotonic zero-tending sequence can be a sequence of rational deviations in the Euclidean norm; characterizing universal pole sequences for approximation by simple partial fractions on compacts in the complex plane; proving the convergence of various greedy algorithms in Banach spaces; improving the known estimates of the convergence rate of greedy algorithms for special dictionaries; description of dictionaries for which m-term approximations coincide with greedy ones; introduction and proof of the convergence of greedy approximation with respect to several dictionaries; results on the weak convergence of consecutive random projections on several convex sets; description of finite-dimensional spaces in which the class of Chebyshev sets coincides with the class of closed monotonically linearly connected sets. All the expected results will be new and should make a significant contribution to approximation theory.

Annotation of the results obtained in 2023
(WN)-property of a Banach space is proved to be responsible for the weak convergence of the greedy algorithm in the class of uniformly smooth spaces. For any norm-reducing set in an Efimov-Stechkin space, it was proved that all possible sums of its elements are dense in this space. All discrete dictionaries in Hilbert space are described, for which Pure Greedy Algorithm and Orthogonal Greedy Algorithm work the same way for every initial element. All three-dimensional normed spaces in which every bounded Chebyshev set is monotonously linearly connected are described. All the results are new, correspond to the world level in the sense that any mathematician would not be ashamed of them, and make a significant contribution to the nonlinear theory of approximations. Three papers were published in peer-reviewed journals, and another paper was accepted for publication.

Publications

1. Bednov B.B. Трехмерные пространства, в которых каждое ограниченное чебышевское множество монотонно линейно связно Математические заметки, Т. 114, вып. 3, стр. 323–338 (year - 2023) https://doi.org/10.4213/mzm13569

2. Borodin P.A. Слабая сходимость жадного алгоритма и WN-свойство Математические заметки, Том 113, выпуск 4, стр. 483–488 (year - 2023) https://doi.org/10.4213/mzm13667

3. Borodin P.A., Burusheva L.Sh. Оценка скорости сходимости дальних проекций на три подпространства Функциональный анализ и его приложения, Т. 57, вып. 2, стр. 100–105 (year - 2023) https://doi.org/10.4213/faa4067

4. K.S.Vishnevetskiy Сравнение чисто жадного и ортогонального жадного алгоритмов Математические заметки, - (year - 2024)

Annotation of the results obtained in 2022
The dichotomy of m-term approximations is established: for any dictionary in a Hilbert space, either the least m-term deviations of any element decrease exponentially, or there are elements with arbitrarily slow decreasing of these deviations. A dictionary is constructed in a separable infinite-dimensional Hilbert space, the m-term deviations with respect to which can form any strictly monotone sequences tending to zero. It is proved that the sequence of rational deviations in the Euclidean norm cannot form an arbitrary strictly monotone sequence tending to zero. The conditions for a subset of the complex plane are found that are necessary or sufficient for the simplest fractions with poles from this set to be able to approximate with any accuracy on an arbitrary compact without holes an arbitrary function holomorphic on this compact. The well-known problem of weak convergence of consecutive random projections on convex sets was investigated, and its connection with the problem of weak convergence of greedy residuals was established when the greedy algorithm acts with respect to several dictionaries. Estimates of the convergence rate of pure greedy algorithm for a dictionary consisting of several linear subspaces in a Hilbert space are obtained. All discrete dictionaries in Hilbert space are described, for which the pure greedy algorithm produces the best m-term approximations. All finite-dimensional normed spaces in which the class of Chebyshev sets coincides with the class of closed monotone path-connected sets are described. All the results are new, correspond to the world level in the sense that any mathematician would not be ashamed of them, and make a significant contribution to the theory of nonlinear approximations.

Publications

1. Bednov B.B. Конечномерные пространства, в которых класс чебышевских множеств совпадает с классом замкнутых и монотонно линейно связных множеств Математические заметки, Том 11, вып. 4, стр.483-493 (year - 2022) https://doi.org/10.4213/mzm13314

2. Borodin P.A. Приближение наипростейшими дробями: универсальные множества полюсов Математические заметки, Матем. заметки, 111:1 (2022), 3–7 (year - 2022) https://doi.org/10.4213/mzm13223

3. Borodin P.A., Kopecka E. Sequences of m-term deviations in Hilbert space Journal of Approximation Theory, V. 284,105821 (year - 2022) https://doi.org/10.1016/j.jat.2022.105821

4. Borodin P.A., Kopecka E. Слабые пределы последовательных проекций и жадных шагов Труды МИАН имени В.А.Стеклова, Т. 319 (year - 2022) https://doi.org/10.4213/tm4264

5. Vishnevetsky K.S. О совпадении чисто жадных и наилучших m-членных приближений Математические заметки, Т. 111, вып. 2, 202-210 (year - 2022) https://doi.org/10.4213/mzm13224

6. Vishnevetsky K.S. Condition of Coincidence between Greedy Approximations and m-Term Ones Moscow University Mathematics Bulletin, 77, 67-72 (year - 2022) https://doi.org/10.3103/S0027132222020085