Dan, Hui; Guo, Kunyu The periodic dilation completeness problem: cyclic vectors in the Hardy space over the infinite-dimensional polydisk. (English) Zbl 1465.42039 J. Lond. Math. Soc., II. Ser. 103, No. 1, 1-34 (2021). Summary: The classical completeness problem raised by A. Beurling [The collected works of Arne Beurling. Volume 1: Complex analysis. Volume 2: Harmonic analysis. Ed. by Lennart Carleson, Paul Malliavin, John Neuberger, John Wermer. Boston etc.: Birkhäuser Verlag (1989; Zbl 0732.01042)] and independently by A. Wintner [Am. J. Math. 66, 564–578 (1944; Zbl 0061.24902)] asks for which \(\psi \in L^2(0,1)\), the dilation system \(\{\psi(kx):k=1,2, \dots\}\) is complete in \(L^2(0,1)\), where \(\psi\) is identified with its extension to an odd 2-periodic function on \(\mathbb{R}\). This difficult problem is nowadays commonly calles as the periodic dilation completeness problem (PDCP). By Beurling’s idea and an application of the Bohr transform, the PDCP is translated as an equivalent problem of characterizing cyclic vectors in the Hardy space \(\mathcal{H}_\infty^2\) over the infinite-dimensional polydisk for coordinate multiplication operators. In this paper, we obtain lots of new results on cyclic vectors in the Hardy space \(\mathcal{H}_\infty^2\). In almost all interesting cases, we obtain sufficient and necessary criterions for characterizing cyclic vectors, and hence in these cases we completely solve the PDCP. Our results cover almost all previous known results on this subject. Cited in 12 Documents MSC: 42C30 Completeness of sets of functions in nontrigonometric harmonic analysis 47A16 Cyclic vectors, hypercyclic and chaotic operators 46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces 46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) Keywords:cyclic vectors in a Hardy space; Bohr transform Citations:Zbl 0732.01042; Zbl 0061.24902 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] N. I.Akhiezer, Lectures on approximation theory, 2nd edn (Nauka, Moscow) (Russian), Approximation theory (Dover, New York, 1992). [2] A.Aleman, J.‐F.Olsen and E.Saksman, ‘Fatou and brothers Riesz theorems in the infinite‐dimensional polydisc’, J. Anal. Math.137 (2019) 429-447. · Zbl 1431.46021 [3] T. 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