# The periodic dilation completeness problem: cyclic vectors in the Hardy space over the infinite‐dimensional polydisk

@article{Dan2020ThePD, title={The periodic dilation completeness problem: cyclic vectors in the Hardy space over the infinite‐dimensional polydisk}, author={Hui Dan and Kunyu Guo}, journal={Journal of the London Mathematical Society}, year={2020}, volume={103} }

The classical completeness problem raised by Beurling and independently by Wintner asks for which ψ∈L2(0,1) , the dilation system {ψ(kx):k=1,2,…} is complete in L2(0,1) , where ψ is identified with its extension to an odd 2‐periodic function on 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…

## 8 Citations

Splitting the Riesz basis condition for systems of dilated functions through Dirichlet series

- MathematicsJournal of Mathematical Analysis and Applications
- 2021

A weighted composition semigroup related to three open problems

- Mathematics
- 2021

The semi-group of weighted composition operators (Wn)n≥1 where Wnf(z) = (1 + z + . . .+ z )f(z) on the classical Hardy-Hilbert space H of the open unit disk is related to the Riemann Hypothesis (RH)…

The Beurling-Wintner problem for step functions

- Mathematics
- 2020

This paper concerns a long-standing problem raised by Beurling and Wintner on completeness of the dilation system {φ(kx) : k = 1, 2, · · · } generated by the odd periodic extension on R of any φ ∈…

Invariant subspaces of weighted Bergman spaces in infinitely many variables

- Mathematics
- 2021

This paper is concerned with polynomially generated multiplier invariant subspaces of the weighted Bergman space Aβ in infinitely many variables. We completely classify these invariant subspaces…

Dirichlet series and the Nevanlinna class in infinitely many variables

- Mathematics
- 2022

Abstract: This paper is concerned with Dirichlet series and holomorphic functions in infinitely many variables. We study the function theory of the Nevanlinna class and the Smirnov class in…

Cyclic Vectors, Outer Functions and Mahler Measure in Two Variables

- MathematicsIntegral Equations and Operator Theory
- 2021

The solutions to the Wintner-Beurling problem in the class of step functions

- Mathematics
- 2020

In this paper, we establish an analytic number theoretic approach toward a classical analysis problem raised by Wintner and Beurling independently in the 1940s. This problem is to seek $\varphi\in…

{f(z)}k∈N in Dirichlet-type spaces

- Mathematics
- 2020

In this paper, we concentrate on power dilation systems {f(z)}k∈N in Dirichlet-type spaces Dt (t ∈ R). When t 6= 0, we prove that {f(z)}k∈N is orthogonal in Dt only if f = cz for some constant c and…

## References

SHOWING 1-10 OF 69 REFERENCES

In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc

- Mathematics
- 2012

— Completeness of a dilation system (φ(nx))n>1 on the standard Lebesgue space L2(0, 1) is considered for 2-periodic functions φ. We show that the problem is equivalent to an open question on cyclic…

Fatou and brothers Riesz theorems in the infinite-dimensional polydisc

- MathematicsJournal d'Analyse Mathématique
- 2019

We study the boundary behavior of functions in the Hardy spaces on the infinite-dimensional polydisc. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several…

SUBMODULES OF THE HARDY MODULE IN INFINITELY MANY VARIABLES

- Mathematics
- 2018

This paper is concerned with polynomially generated submodules of the Hardy module H2(D∞ 2 ). Since the polynomial ring P∞ in infinitely many variables is not Noetherian, some standard tricks for…

A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$

- Mathematics
- 1995

For a function $\varphi$ in $L^2(0,1)$, extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates $\varphi(nx)$, $n=1,2,3,\ldots$, constitutes a…

A correction to “In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc”

- Mathematics
- 2018

— This note corrects some inaccuracies remarked in the paper mentioned in the title. It contains also a few references to recent developments on the dilations f(nx) completeness problem and points…

Dilation theory and analytic model theory for doubly commuting sequences of $C_{.0}$-contractions

- Mathematics
- 2019

Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilation to a Hardy shift and thus established an elegant analytic functional model for contractions of class $C_{\cdot0}$. This has…

On the shift semigroup on the Hardy space of Dirichlet series

- Mathematics
- 2010

AbstractWe develop a Wold decomposition for the shift semigroup on the Hardy space $$
\mathcal{H}^2
$$ of square summable Dirichlet series convergent in the half-plane $$
\Re (s) > 1/2
$$. As an…

Systems of dilated functions: Completeness, minimality, basisness

- Mathematics
- 2016

The completeness, minimality, and basis property in L2[0, π] and Lp[0, π], p ≠ 2, are considered for systems of dilated functions un(x) = S(nx), n ∈ N, where S is the trigonometric polynomial S(x) =…

A Hardy space analysis of the Báez-Duarte criterion for the RH

- MathematicsAdvances in Mathematics
- 2019

Introduction to analytic number theory

- Mathematics
- 1976

This is the first volume of a two-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years. It provides an introduction…