The classical completeness problem raised by Beurling and independently by Wintner 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 called 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 ${\mathbf{H}}_{\infty }^2$ 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 ${\mathbf{H}}_{\infty }^2$. 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.

中文翻译：

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周期膨胀完整性问题：无限维多圆盘上Hardy空间中的循环矢量

Beurling提出并由Wintner独立提出的经典完整性问题要求 $\psi \in {\mathrm{大号}}^2（0，\mathrm{1个}）$，扩张系统 $\{\psi （ķX）：ķ=\mathrm{1个}，2，\dots \}$ 在完成 ${\mathrm{大号}}^2（0，\mathrm{1个}）$，在哪里 $\psi$ 被标识为其扩展到一个奇2周期函数 $\mathbb{[R}$。如今，这个难题被普遍称为周期性膨胀完整性问题（PDCP）。通过Beurling的想法和Bohr变换的应用，将PDCP转换为表征Hardy空间中循环矢量的等效问题${\mathbf{H}}_{\infty }^2$在无限维多磁盘上用于坐标乘法运算符。在本文中，我们获得了有关Hardy空间中循环矢量的许多新结果${\mathbf{H}}_{\infty }^2$。在几乎所有有趣的情况下，我们都获得了足够的和必要的准则来表征循环矢量，因此在这些情况下，我们可以完全解决PDCP。我们的结果涵盖了该主题几乎所有以前已知的结果。