On the growth of Lebesgue constants for convex polyhedra

Abstract

In the paper, new estimates of the Lebesgue constant $$\mathcal{L}(W)=\frac{1}{(2\pi {)}^d}{\int }_{{\mathbb{T}}^d}\left|\sum _{k\in W\cap {\mathbb{Z}}^d}{e}^{i(k,x)}\right|\mathrm{d}x$$ for convex polyhedra $W\subset {\mathbb{R}}^d$ are obtained. The main result states that if $W$ is a convex polyhedron such that $[0,m_1]\times \cdots \times [0,m_d]\subset W\subset [0,n_1]\times \cdots \times [0,n_d]$, then $$c(d)\prod _{j=1}^d\log (m_j+1)\le \mathcal{L}(W)\le C(d)s\prod _{j=1}^d\log (n_j+1),$$ where $s$ is a size of the triangulation of $W$.

accepted in Trans. Amer. Math. Soc