Multi-dimensional Approximations and Their Applications

• We have proposed a unified method for analyzing greedy algorithms both in Hilbert and in Banach spaces. We have determined and analyzed a new class of algorithms – weak biorthogonal greedy algorithms that includes as a particular case such important algorithms as the Weak Chebyshev Greedy Algorithm and the Greedy Algorithm with Free Relaxation. We have determined and investigated a new useful algorithm – the scaled weak relaxation algorithm. We have produced theorems of convergence and evaluation of convergence rate of such algorithms.
• We have investigated efficiency of greedy algorithms in respect of special dictionaries. We have produced Lebesgue inequalities for dictionaries with special structure starting with different types of base and ending with overflown dictionaries that meet a more general requirement than the requirement of limited isometry.
• We have proven existence of special sets of points that provide discretization of norms of trigonometric polynomials with harmonics from parallelepipeds of different shapes. We have shown the built sets are optimal in terms of numbers of points. Our researchers have proposed a general method for building grids for good (optimal with up to log precision in respect of quantity) discretization of elements of finite-dimensional subspace. We have investigated dispersion of grids (Fibonacci and Frolov). It has been proven that these sets have the optimal dispersion in terms of order. We have introduced and thoroughly studied the notion of fixed volume discrepancy (which is more specific compared to standard dispersion). We have compiled a work in which we investigate connection between such fundamental notions as numerical integration, discrepancy, and nonlinear approximations.
• Our researchers have proven that there exists a sequence of nontrivial three-dimensional polynomials represented in the complex form with natural coefficients converging to zero almost everywhere; we have proven that a function can be found for witch linear combinations of shifts with unitary coffieicents are dense in spaces integrated in the power of p not less than 2. We have found lower estimates of L-norms of exponential sums.
• Our Laboratory has studied simplest fractions (logarithmic derivatives of polynomials) with poles on the boundaries of unlimited simply-connected region of D-complex plane in the A(D) function space homomorphic within D. We have found different conditions necessary or sufficient for this density. It has been have proven that in the case of the D band simplest fractions with poles on the boundaries of of D are dense in A(D) and produced estimate of rate of approaching of concrete functions on compacts within the band. A corresponding publication is ready for publishing.
• We have investigated some qualities of smooth sums of ridge functions (smooth waves) determined on a convex body. We have investigated qualities of smoothness of reviewed sums if Dini conditions are met for modules of continuity of n-th order partial derivatives.
• Our researchers have found equivalence of important problems of approximation of the low rank upper triangular matrix with ones above the diagonal and finding Kolmogorov diameter of an oblique tetrahedron. We have researched special random method of approximation for this problem for which we found an estimate of error that are precise in terms of order.

Education and career development:

• One candidate dissertation has been defended.
• We have read new special courses «Convex geometry and problems of compressed dimensions and problems of compressed dimensions» and «Approximations theory of its applications» for students of the Faculty of Mechanics and Mathematics of the Moscow State University.
• Our has organized two seminars on the grounds of the Laboratory and conducted a number of meetings within these seminars.
• We have conducted the International conference «High-dimensional approximation and discretization» (23-29 September 2018)with about 50 participants from Germany, France, Austria, the United Kingdom, and Kazakhstan).

Collaborations:

Big Data Storage and Analysis Center of the Moscow State University (Russia), University of South Carolina (USA): joint research