On a problem of Kahane in higher dimensions

TL;DR

This paper characterizes real analytic mappings between tori that transform absolutely convergent Fourier series into uniformly convergent ones, exploring differences between rectangular and square summation methods.

Contribution

It provides a comprehensive characterization of such mappings and compares convergence behaviors under different summation techniques.

Findings

Characterization of mappings transforming absolutely convergent Fourier series to uniformly convergent series.

Differences in convergence properties between rectangular and square summation methods.

Insights into the structure of real analytic mappings on tori.

Abstract

We characterise those real analytic mappings between any pair of tori which carry absolutely convergent Fourier series to uniformly convergent Fourier series via composition. We do this with respect to rectangular summation. We also investigate uniform convergence with respect to square sums and highlight the differences which arise.