Estimates of Lebesgue constants via Fourier transforms. Many dimensions

TL;DR

This paper surveys methods for estimating Lebesgue constants of multiple Fourier series using Fourier transforms, including well-known, lesser-known, and new unpublished results across various dimensions.

Contribution

It provides a comprehensive overview of techniques for bounding Lebesgue constants via Fourier transform estimates, including novel unpublished findings.

Findings

Compilation of known and new estimates of Lebesgue constants

Illustration of techniques using Fourier transforms

Extension to multiple dimensions

Abstract

This is an attempt of a comprehensive survey of the results in which estimates of the norms of linear means of multiple Fourier series, the Lebesgue constants, are obtained by means of estimating the Fourier transform of a function generating such a method. Only few proofs are given in order to illustrate a general idea of techniques applied. Among the results are well known elsewhere as well as less known or published in an unacceptable journals and several new unpublished results.