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 belong 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 ∈ \(U(\mathbb{T})\) whenever f ∈ \(U(\mathbb{T})\). We present certain sufficient conditions for a function to be a multiplier and also obtain some Salem-type results.