On the growth of Lebesgue constants for convex polyhedra

TL;DR

This paper provides new bounds on the Lebesgue constants for convex polyhedra, showing how they grow with the polyhedron's size and triangulation complexity, which advances understanding in harmonic analysis.

Contribution

The paper introduces new estimates for Lebesgue constants of convex polyhedra, relating their growth to geometric parameters and triangulation complexity.

Findings

Lebesgue constants grow logarithmically with polyhedron size.

Bounds depend on the polyhedron's triangulation size.

Provides explicit inequalities for Lebesgue constants in terms of geometric parameters.

Abstract

In the paper, new estimates of the Lebesgue constant $$ \mathcal{L}(W)=\frac1{(2\pi)^d}\int_{\mathbb{T}^d}\bigg|\sum_{{k}\in W\cap \mathbb{Z}^d} e^{i({k},\,{x})}\bigg| {\rm d}{ x} $$ for convex polyhedra $W\subset\mathbb{R}^d$ are obtained. The main result states that if $W$ is a convex polyhedron such that $[0,m_1]\times\dots\times [0,m_d]\subset W\subset [0,n_1]\times\dots\times [0,n_d]$, 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 $W$.