On the growth of Lebesgue constants for convex polyhedra

Abstract

In this paper, new estimates of the Lebesgue constant L ( W ) = 1 ( 2 π ) d ∫ T d | ∑ k ∈ W ∩ Z d e i ( k , x ) | d x \begin{equation*} \mathcal {L}(W)=\frac 1{(2\pi )^d}\int _{{\Bbb T}^d}\bigg |\sum _{\mathbfit {k}\in W\cap {\Bbb Z}^d} e^{i(\mathbfit {k}, \mathbfit {x})}\bigg | \textrm {d}\mathbfit {x} \end{equation*} for convex polyhedra W ⊂ R d W\subset {\Bbb R}^d are obtained. The main result states that if W W is a convex polyhedron such that [ 0 , m 1 ] × ⋯ × [ 0 , m d ] ⊂ W ⊂ [ 0 , n 1 ] × ⋯ × [ 0 , n d ] [0,m_1]\times \dots \times [0,m_d]\subset W\subset [0,n_1]\times \dots \times [0,n_d] , then c ( d ) ∏ j = 1 d log ⁡ ( m j + 1 ) ≤ L ( W ) ≤ C ( d ) s ∏ j = 1 d log ⁡ ( n j + 1 ) , \begin{equation*} 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), \end{equation*} where s s is a size of the triangulation of W W .