Discrete imbedding theorems and Lebesgue constants

Abstract

The order of growth of the Lebesgue constant for a “hyperbolic cross” is found:

$$L_R = \smallint _{T^2 } \left| {\sum\nolimits_{0< \left| {v_1 v_2 } \right| \leqslant R^2 } {e^{2\pi ivx} } } \right|dx\begin{array}{*{20}c} \smile \\ \frown \\ \end{array} R^{1/_2 } , R \to \infty $$

. Estimates are obtained by applying a discrete imbedding theorem. It is proved that among all convex domains in E², the square gives rise to a Lebesgue constant with the slowest growth ln²R.