IRP on Modern Trends in Fourier Analysis

A great number of topics in discrete geometry are naturally related to or even directly arise from questions of approximation theory. Such topics include, in particular, lattices, discrepancy theory, uniform distribution, distribution of points/electrons on spheres and other manifolds (these subjects are closely linked to numerical integration and cubature formulas), covering and packing problems, metric entropy (the relation to approximation theory is self-evident in this case), triangulations and partitions (connected to the finite element method), as well as many other areas, e.g. chromatic numbers, coding theory, computational geometry, some aspects of convex geometry etc.

These questions have been extremely vital in the last few years. Many important contributions have been recently made in various directions, e.g. the solution of the Korevaar–Meyer conjecture on the optimal size of spherical designs, new world record in counterexamples to Borsuk’s conjecture, improved lower bounds for the discrepancy function in higher dimensions, the proof that acute triangualtions of a cube exist only in dimensions 2 and 3, to name just a few. Hence the time is ripe to dedicate a part of the program to these questions.

While at the first glance the reader may wonder why analysis is mentioned in the name of the topic, a deeper look into the field reveals that almost all important problems in the subject utilize, in one way or another, on various methods and techniques arising from harmonic, Fourier, complex, geometric, or functional analysis. Indeed, Fourier analysis has been always used to study the behavior of lattices, the problems of positioning electrons on manifolds naturally lead to potential analysis, discrepancy theory uses many ideas from classical and modern harmonic analysis and approximation theory (e.g. wavelets, hyperbolic-cross approximations), Brouwertype fixed-point theorems have been recently successfully applied to demonstrate the existence of spherical designs and to other problems of approximation theory, convex geometry heavily relies on Fourier and functional analysis. Therefore this choice of topic is not only perfectly appropriate, but is almost uniquely tailored to a program focused on analysis and approximation theory.

A small sample of a variety of interesting problems in this field could include the following. The problem of covering is a classical problem in discrete computational geometry. Establishing covering numbers for various sets, e.g., unit balls of Banach spaces leads to far-reaching consequences in approximation theory. A fine example of a covering problem is the famous Borsuk’s conjecture, which states that a body in Rd of diameter 1 can be covered by at most d + 1 bodies of diameter strictly less than 1. It has been disproved a while ago, but the smallest dimension in which it fails is still unknown (very recently it was shown that it fails in dimension 64). A closely related question deals with covering of a unit ball with balls of radius smaller than (but close to) 1. The right bounds in this old open problem are still not known. Covering results are in turn related to optimal spherical codes and construction of incoherent systems of large cardinality, which are needed, in particular, in compressed sensing. In another vein, discrepancy theory studies approximations of continuous objects by discrete. The exact behavior of the discrepancy of a finite point set with respect to axis parallel rectangles is still mysterious in dimensions higher than 2. This problem is directly related to numerical integration of functions with mixed smoothness, but is also linked to covering properties of such classes. Numerous other problems of discrepancy theory, especially in higher dimensions, are simultaneously interesting and challenging. In addition, problems of deterministic constructions of low-discrepancy sets, optimal spherical designs and spherical codes (whose existence is often established non-constructively) are important directions in constructive approximation with useful applications.

This topic will present an excellent opportunity to widen the scope of the program and bring in participants from other communities, introducing new people and new subjects to the approximation theory and analysis groups, which would inevitably lead to cross-fertilization and exchange of ideas, and will no doubt enrich both fields.