## Monday January 31, 2005, 4 PM, Goldsmith 317

### tea at
3:30 PM at the Math Department lounge (Goldsmith 300)

**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 *L*^{1} 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.