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

Volume 247 / 2019

Vladimir Lebedev Studia Mathematica 247 (2019), 273-283 MSC: Primary 42B35; Secondary 42B05, 42A20. DOI: 10.4064/sm170714-16-3 Published online: 31 January 2019


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 $\varphi$ of a certain special form on $\mathbb T^d, \,d\geq 2,$ we obtain lower bounds for the norms $\|e^{i\lambda\varphi}\|_A$ as $\lambda\rightarrow\infty$. In particular, we show that if $\varphi(x, y)=a(x)|y|$ for $|y|\leq\pi$, where $a\in A(\mathbb{T})$ is a nonconstant real function, then $\|e^{i\lambda\varphi}\|_{A(\mathbb{T}^2)}\gtrsim |\lambda|$.


  • Vladimir LebedevNational Research University Higher School of Economics
    34 Tallinskaya St.
    Moscow 123458, Russia

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