A Beurling-Helson type theorem for modulation spaces

定量Beurling-Helel定理

The quantitative Beurling-Helson Theorem

本文研究了函数ϕ在特定条件下的形式，证明了当ϕ满足特定范数条件时，ϕ可表示为整数倍加常数的形式。

Quantitative version of Beurling-Helson theorem

It is proved that any continuous function f on the unit circle such that the sequence e^{in f}, n=1,2,... has small Wiener norm \| e^{in f} \|_A = o (\frac{\log^{1/22} |n|}{(\log \log |n|)^{3/11}}), is linear. Moreover, we get lower bounds for Wiener norm of characteristic functions of subsets from Z_p in the case of prime p.

Quantitative estimates in Beurling--Helson type theorems

We consider the spaces $A_p(\mathbb T)$ of functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\fu{\f}=\{\fu{\f}(k), ~k \in \mathbb Z\}$ belongs to $l^p, ~1\leq p<2$. The norm on $A_p(\mathbb T)$ is defined by $\|f\|_{A_p}=\|\fu{\f}\nolinebreak\|_{l^p}$. We study the rate of growth of the norms $\|e^{iλφ}\|_{A_p}$ as $|λ|\rightarrow \infty, ~λ\in\mathbb R,$ for $C^1$ -smooth real functions $φ$ on $\mathbb T$. The results have natural applications to the problem on changes of variable in the spaces $A_p(\mathbb T)$.

Quantitative aspects of the Beurling--Helson theorem: Phase functions of a special form

We consider the space $A(\mathbb{T}^d)$ of absolutely convergent Fourier series on the torus $\mathbb{T}^d$. The norm on $A(\mathbb{T}^d)$ is naturally defined by $\|f\|_{A}=\|\widehat{f}\|_{l^1}$, where $\widehat{f}$ is the Fourier transform of a function $f$. For real functions $φ$ of a certain special form on $\mathbb T^d, \,d\geq 2,$ we obtain lower bounds for the norms $\|e^{iλφ}\|_A$ as $λ\rightarrow\infty$. In particular, we show that if $φ(x, y)=a(x)|y|$ for $|y|\leqπ$, where $a\in A(\mathbb{T})$ is an arbitrary nonconstant real function, then $\|e^{iλφ}\|_{A(\mathbb{T}^2)}\gtrsim |λ|$…

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

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 $\|f\|_{A(\mathbb T)}=\|\hat{f}\|_{l^1(\mathbb Z)}$. According to the known Beurling--Helson theorem, if $ϕ: \mathbb T\rightarrow\mathbb T$ is a continuous mapping such that $\|e^{inϕ}\|_{A(\mathbb T)}=O(1), ~n\in\mathbb Z,$ then $ϕ$ is linear. It was conjectured by Kahane that the same conclusion about $ϕ$ is true under the assumption that $\…

Estimates in Beurling--Helson type theorems. Multidimensional case

We consider the spaces $A_p(\mathbb T^m)$ of functions $f$ on the $m$ -dimensional torus $\mathbb T^m$ such that the sequence of the Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z^m\}$ belongs to $l^p(\mathbb Z^m), ~1\leq p<2$. The norm on $A_p(\mathbb T^m)$ is defined by $\|f\|_{A_p(\mathbb T^m)}=\|\hat{f}\|_{l^p(\mathbb Z^m)}$. We study the rate of growth of the norms $\|e^{iλφ}\|_{A_p(\mathbb T^m)}$ as $|λ|\rightarrow \infty, ~λ\in\mathbb R,$ for $C^1$ -smooth real functions $φ$ on $\mathbb T^m$ (the one-dimensional case was investigated by the author earlier). The lower estim…