Lebesgue constants of multiple Fourier series

This is an attempt of a comprehensive treatment of the results concerning estimates of the \( L^1 \)-norms of linear means of multiple Fourier series, the Lebesgue constants. Most of them are obtained by estimating the Fourier transform of a function generating such a method. Frequently the properties of the support of this function affect distinctive features in behavior of these norms. By this geometry enters and works hand-in-hand with analysis; moreover, the results are classified mostly in accordance with their geometrical nature. Not rarely Number Theory tools are brought in. We deal only with the trigonometric case – no generalizations for other orthogonal systems are discussed nor are applications to approximation. Several open problems are posed.