Ingham type inequalities in lattices

TL;DR

This paper extends Ingham's inequalities to all regular two-dimensional lattices, identifying optimal connected domains for integration, thus advancing understanding of exponential systems in multi-dimensional harmonic analysis.

Contribution

It determines the optimal connected integration domains for Ingham inequalities in all regular two-dimensional lattices, filling a gap in multi-dimensional harmonic analysis.

Findings

Identified optimal connected domains for 2D lattices

Extended Ingham inequalities to all regular 2D lattices

Enhanced understanding of exponential systems in multi-dimensional spaces

Abstract

A classical theorem of Ingham extended Parseval's formula of the trigonometrical system to arbitrary families of exponentials satisfying a uniform gap condition. Later his result was extended to several dimensions, but the optimal integration domains have only been determined in very few cases. The purpose of this paper is to determine the optimal connected integration domains for all regular two-dimensional lattices.