Periodic Reporting for period 1 - HDAD (High Dimensional Approximation and Discretization)
Berichtszeitraum: 2023-09-01 bis 2025-08-31
The project addresses two fundamental problems in modern approximation theory.
The first concerns the sampling discretization of integral norms, a central question that investigates how integral norms on function classes can be approximated by evaluating the functions at a fixed and relatively small set of points. In many applications, particularly in high-dimensional settings, the computation of integral norms is highly complex and demands substantial analytical and computational resources. Consequently, discretization problems are closely related to numerical integration and information-based complexity. However, unlike classical problems of numerical integration, sampling discretization problems belong to the domain of nonlinear approximation theory. One of the main objectives in this area is to determine the smallest integer m for which the initial measure can be replaced by a discrete uniform measure supported on m points, ensuring that the corresponding continuous and discrete norms remain comparable.
The second problem focuses on the properties of integral norms of polynomials on convex domains and more general settings. In particular, for a given measure, Markov-Bernstein-type inequalities provide estimates for the integral L^p-norm of the gradient of a polynomial in terms of the L^p-norm of the polynomial itself. These inequalities serve as classical tools for studying polynomial approximation in high-dimensional spaces through reverse Jackson-type theorems and play an important role in the discretization of integral norms on spaces of polynomials.
The first concerns the sampling discretization of integral norms, a central question that investigates how integral norms on function classes can be approximated by evaluating the functions at a fixed and relatively small set of points. In many applications, particularly in high-dimensional settings, the computation of integral norms is highly complex and demands substantial analytical and computational resources. Consequently, discretization problems are closely related to numerical integration and information-based complexity. However, unlike classical problems of numerical integration, sampling discretization problems belong to the domain of nonlinear approximation theory. One of the main objectives in this area is to determine the smallest integer m for which the initial measure can be replaced by a discrete uniform measure supported on m points, ensuring that the corresponding continuous and discrete norms remain comparable.
The second problem focuses on the properties of integral norms of polynomials on convex domains and more general settings. In particular, for a given measure, Markov-Bernstein-type inequalities provide estimates for the integral L^p-norm of the gradient of a polynomial in terms of the L^p-norm of the polynomial itself. These inequalities serve as classical tools for studying polynomial approximation in high-dimensional spaces through reverse Jackson-type theorems and play an important role in the discretization of integral norms on spaces of polynomials.
First, we studied the problem of sampling discretization for an arbitrary compact class of continuous functions, relating the discretization error to the decay of entropy numbers in the uniform norm. Second, we investigated integral norms sampling discretization in the setting where the L^p-norm is replaced by an arbitrary Orlicz norm. For randomly chosen points, we obtained near-optimal estimates for the sufficient number of samples, coinciding with the known results in the L^p-case. Moreover, these results were applied to derive new estimates for the error in the sampling recovery problem. Third, in the classical L^p-case for p>2, we established new estimates for the number of points sufficient for accurate discretization, significantly improving previously known results. Finally, we derived new Markov–Bernstein-type inequalities for polynomials on the unit cube.
The project has produced several substantial theoretical results that advance the understanding of sampling discretization and polynomial inequalities in high-dimensional analysis. New estimates were obtained for the sufficient number of sampling points in the problem of discretization of integral norms, extending the classical framework to Orlicz spaces and linking the discretization error to the decay of entropy numbers. In the classical case, improved upper bounds for the number of sampling points were established, significantly strengthening previously known results. The study also led to new Markov–Bernstein-type inequalities for polynomials on the unit cube.
The potential impact of these results lies in their broad applicability to approximation theory, information-based complexity, and numerical analysis. The new methods and inequalities developed within the project can serve as a foundation for efficient sampling and recovery techniques in high-dimensional settings.
To ensure further uptake and success, future work should focus on:
1) extending these theoretical frameworks to broader settings;
2) exploring interdisciplinary applications in applied mathematics, machine learning, and signal processing;
3) strengthening international collaboration and dissemination to the Information-Based Complexity community.
The potential impact of these results lies in their broad applicability to approximation theory, information-based complexity, and numerical analysis. The new methods and inequalities developed within the project can serve as a foundation for efficient sampling and recovery techniques in high-dimensional settings.
To ensure further uptake and success, future work should focus on:
1) extending these theoretical frameworks to broader settings;
2) exploring interdisciplinary applications in applied mathematics, machine learning, and signal processing;
3) strengthening international collaboration and dissemination to the Information-Based Complexity community.