Un th\'eor\`eme de Helson pour des s\'eries de Walsh

TL;DR

This paper extends Helson and Katznelson's theorems to Walsh series, showing that positive partial sums imply coefficients tend to zero, but not necessarily forming a Fourier-Lebesgue series.

Contribution

It generalizes classical results from trigonometric to Walsh series, highlighting differences in convergence properties.

Findings

Coefficients tend to zero when partial sums are positive

Walsh series can behave differently from Fourier-Lebesgue series

Extension of Helson's theorem to Walsh context

Abstract

Extension to Walsh series of theorems of Helson and Katznelson on trigonometric series, saying that a trigonometric series whose partial sums are positive has its coefficients tend to zero but is not necessarily a Fourier-Lebesgue series