Uniformly convergent Fourier series and multiplication of functions

TL;DR

This paper investigates the conditions under which functions act as pointwise multipliers for the space of continuous functions on the circle with uniformly convergent Fourier series, extending Salem's results.

Contribution

It provides sufficient conditions for functions to serve as multipliers of the space $U( ext{T})$ and extends Salem's classical results on Fourier series convergence.

Findings

Identifies conditions for functions to be multipliers of $U( ext{T})$

Shows that the product of two functions in $U( ext{T})$ may not stay in $U( ext{T})$

Extends Salem's results on Fourier series convergence

Abstract

Let $U(\mathbb T)$ be the space of all continuous functions on the circle $\mathbb T$ whose Fourier series converges uniformly. Salem's well-known example shows that a product of two functions in $U(\mathbb T)$ does not always belongs to $U(\mathbb T)$ even if one of the factors belongs to the Wiener algebra $A(\mathbb T)$. In this paper we consider pointwise multipliers of the space $U(\mathbb T)$, i.e., the functions $m$ such that $mf\in U(\mathbb T)$ whenever $f\in U(\mathbb T)$. We present certain sufficient conditions for a function to be a multiplier and also obtain some results of Salem type.