Uniformly convergent Fourier series and multiplication of functions

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.

References in corpus (1)

• On uniform convergence of Fourier series