Speaker: Elijah Liflyand (Bar Ilan University)
Title: Lebesgue constants of multiple Fourier series
Abstract: This is an attempt of a comprehensive treatment of the results concerning estimates of the L1 norms of trigonometric sums (Dirichlet kernels). These norms are commonly referred to as Lebesgue constants and are the norms of operators of taking partial Fourier sums. In the one-dimensional case there is a natural way of summing the Fourier coefficients and, consequently, there exist explicit formulas for Dirichlet kernels and exact asymptotic formulas for the Lebesgue constants. No natural ordering of Fourier coefficients of multiple Fourier series exists, and hence various ways of ordering are of considerable interest and importance in theory and applications.
In the multivariate case various ways of summing are generated by certain sets. These lead to rectangular, cubic, polyhedral, spherical, hyperbolic, etc., partial Fourier sums, and consequently to different estimates of the corresponding Lebesgue constants. By this geometry enters and works hand-in-hand with analysis; moreover, the results are classified mostly in accordance with their geometrical nature. Most of these results are obtained by estimating the Fourier transform of the indicator function of the set generating partial sums.
Frequently the properties of the generating set affects distinctive features in the behaviour of these norms. The rate of the tendency of these norms to infinity gives a measure of divergence of Fourier series. Not rarely Number Theory tools are brought in. We deal only with the trigonometric case Ð no generalizations to other orthogonal systems are discussed. Several open problems are posed.