Absolutely convergent Fourier series. An improvement of the Beurling--Helson theorem

Abstract

We consider the space $A(\mathbb{T})$ of all continuous functions $f$ on the circle $\mathbb{T}$ such that the sequence of Fourier coefficients $\hat{f}=\{\hat{f}(k),k\in \mathbb{Z}\}$ belongs to $l^1(\mathbb{Z})$. The norm on $A(\mathbb{T})$ is defined by $\Vert f{\Vert }_{A(\mathbb{T})}=\Vert \hat{f}{\Vert }_{l^1(\mathbb{Z})}$. According to the known Beurling--Helson theorem, if $\varphi :\mathbb{T}\to \mathbb{T}$ is a continuous mapping such that $\Vert e^{in\varphi }{\Vert }_{A(\mathbb{T})}=O(1),n\in \mathbb{Z},$ then $\varphi$ is linear. It was conjectured by Kahane that the same conclusion about $\varphi$ is true under the assumption that $\Vert e^{in\varphi }{\Vert }_{A(\mathbb{T})}=o(\log |n|)$. We show that if $\Vert e^{in\varphi }{\Vert }_{A(\mathbb{T})}=o((\log \log |n|/\log \log \log |n|{)}^{1/12})$ then $\varphi$ is linear.

A typo on page 14 line 4 is corrected. A typo on page 16 line 5 is corrected. Notation in the proof of Lemma 2 are replaced with more readable ones

References in corpus (2)

• The Littlewood-Gowers problem
• Estimates in Beurling--Helson type theorems. Multidimensional case

Cited by in corpus (2)

• Homeomorphic Changes of Variable and Fourier Multipliers
• Quantitative aspects of the Beurling--Helson theorem: Phase functions of a special form