Beurling's Theorem Research Articles

Continuity of inner-outer factorization and cross sections from invariant subspaces to inner functions

Let H∞ be the Banach algebra of bounded analytic functions on the unit open disc D equipped with the supremum norm. As well known, inner functions play an important role in the study of bounded analytic functions. In this paper, we are interested in the study of inner functions. Following by the canonical inner-outer factorization decomposition, define Qinn and Qout the maps from H∞ to I the set of inner functions and F the set of outer functions, respectively. In this paper, we study the H2-norm continuity and H∞-norm discontinuity of Qinn and Qout on some subsets of H∞. On the other hand, the Beurling theorem connects inner functions and invariant subspaces of the multiplication operator Mz. We show the nonexistence of continuous cross section from some certain invariant subspaces to inner functions in the supremum norm. The continuity problem of Qinn and Qout on Hol(D‾), the set of all analytic functions in the closed unit disk, are considered. In addition, we also study the factor maps Q[inn] and Q[out] from H∞ to I/T and F/T, where I/T is the quotient space of I mod T and F/T is the quotient space of F mod T.

Octonion Beurling theorem: Uncertainty principles for Hermite functions on ℝ3

We generalize Beurling's theorem to the octonion Fourier transform for octonion‐valued functions. Despite challenges such as the failure of the octonion Fourier transform to change differentiation into multiplication and the absence of the octonionic Plancherel's theorem, we establish the octonionic Beurling theorem for Hermite functions on by introducing an integral condition involving the octonion Fourier transform. This extension of uncertainty principles by Hardy, Gelfand–Shilov, and Cowling–Price to the octonionic setting demonstrates that the function and its Fourier transform cannot have arbitrary Gaussian decay simultaneously.

Beurling's theorem for the Clifford‐Fourier transform

We generalize Beurling's theorem for the Clifford‐Fourier transform and give some of its applications. We deduce analogs of Hardy, Cowling‐Price, and Gelfand‐Shilov theorems in the Clifford analysis setting.

Dilation theory and analytic model theory for doubly commuting sequences of C·0-contractions

It is known that every C·0-contraction has a dilation to a Hardy shift. This leads to an elegant analytic functional model for C·0-contractions, and has motivated lots of further works on the model theory and generalizations to commuting tuples of C·0-contractions. In this paper, we focus on doubly commuting sequences of C·0-contractions, and establish the dilation theory and the analytic model theory for these sequences of operators. These results are applied to generalize the Beurling-Lax theorem and Jordan blocks in the multivariable operator theory to the operator theory in the infinite-variable setting.

Oscillation of the Remainder Term in the Prime Number Theorem of Beurling, “Caused by a Given ζ-Zero”

Abstract Continuing previous studies of the Beurling zeta function, here, we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the question of Littlewood, who asked for explicit oscillation results provided a zeta-zero is known. We prove that given a zero $\rho _0$ of the Beurling zeta function $\zeta _{{\mathcal {P}}}$ for a given number system generated by the primes ${\mathcal {P}}$, the corresponding error term $\Delta (x):=\psi _{{\mathcal {P}}}(x)-x$, where $\psi _{{\mathcal {P}}}(x)$ is the von Mangoldt summatory function shows oscillation in any large enough interval, as large as $\frac {\pi /2-\varepsilon }{|\rho _0|}x^{\Re \rho _0}$. The somewhat mysterious appearance of the constant $\pi /2$ is explained in the study. Finally, we prove as the next main result of the paper the following: given $\varepsilon>0$, there exists a Beurling number system with primes ${\mathcal {P}}$, such that $|\Delta (x)| \le \frac {\pi /2+\varepsilon }{|\rho _0|}x^{\Re \rho _0}$. In this 2nd part, a nontrivial construction of a low norm sine polynomial is coupled by the application of the wonderful recent prime random approximation result of Broucke and Vindas, who sharpened the breakthrough probabilistic construction due to Diamond, Montgomery, and Vorhauer.

Beurling’s theorem on the Heisenberg group

We formulate and prove an analogue of Beurling's theorem for the Fourier transform on the Heisenberg group. As a consequence we deduce Hardy and Cowling-Price theorems.

Analogues of Beurling's theorem for some integral transforms

A famous result of Beurling says that if the Fourier transform of a non-zero integrable function on the real line has certain exponential decay, then the function cannot vanish on a set of positive Lebesgue measure. We prove an analogue of Beurling's theorem for Hankel transform and some several variable analogues of Beurling's theorem for Fourier transform as a consequence of similar results for Dunkl transform. We also prove an analogue of Beurling's theorem for spectral projections associated to the Dunkl-Laplacian.

A Beurling theorem for almost-invariant subspaces of the shift operator

A complete characterization of nearly-invariant subspaces of finite defect for the backward shift operator acting on the Hardy space is provided in the spirit of Hitt and Sarason's theorems. As a corollary we describe the almost-invariant subspaces for the shift and its adjoint.

A Generalized Beurling Theorem for Some Lie Groups

For any real numbers p,q ≥ 1, we present in this paper a (p, q)-generalized version of Beurling’s uncertainty principle for ℝn, which largely extends the classical Beurling’s theorem. We then define its analog for compact extensions of ℝn and also for Heisenberg groups.

A Beurling theorem for noncommutative Hardy spaces associated with semifinite von Neumann algebras with unitarily invariant norms

We prove a Beurling-type theorem for H∞-invariant spaces of Lα(M,τ), where α is a unitarily invariant, locally ∥⋅∥1-dominating, mutually continuous norm with respect to τ, where M is a von Neumann algebra with a faithful, normal, semifinite tracial weight τ, and H∞ is an extension of Arveson's noncommutative Hardy space. We use our main result to characterize the H∞-invariant subspaces of a noncommutative Banach function space I(τ) with the norm ∥⋅∥E on M, the crossed product of a semifinite von Neumann algebra by an action β, and B(H) for a separable Hilbert space H.

Beurling’s theorem for the quaternion Fourier transform

The two-sided quaternion Fourier transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization of Beurling's theorem, Hardy, Cowling-Price and Gelfand-Shilov theorems, is obtained for the two-sided quaternion Fourier transform.

A Beurling Theorem for Generalized Hardy Spaces on a Multiply Connected Domain

AbstractThe object of this paper is to prove a version of the Beurling–Helson–Lowdenslager invariant subspace theorem for operators on certain Banach spaces of functions on a multiply connected domain in ℂ. The norms for these spaces are either the usual Lebesgue and Hardy space norms or certain continuous gauge norms. In the Hardy space case the expected corollaries include the characterization of the cyclic vectors as the outer functions in this context, a demonstration that the set of analytic multiplication operators is maximal abelian and reflexive, and a determination of the closed operators that commute with all analytic multiplication operators.

A General Vector-Valued Beurling Theorem

Suppose \(\alpha \) is a rotationally symmetric norm on \(L^{\infty }\left( \mathbb {T}\right) \) and \(\beta \) is a “nice” norm on \(L^{\infty }\left( \Omega ,\mu \right) \) where \(\mu \) is a \(\sigma \)-finite measure on \(\Omega \). We prove a version of Beurling’s invariant subspace theorem for the space \(L^{\beta }\left( \mu ,H^{\alpha }\right) .\) Our proof uses the version of Beurling’s theorem on \(H^{\alpha }\left( \mathbb {T}\right) \) in Chen (Adv Appl Math, 2016) and measurable cross-section techniques. Our result significantly extends a result of Rezaei, Talebzadeh, and Shin (Int J Math Anal 6:701–707, 2012).

Cyclicity of reproducing kernels in weighted Hardy spaces over the bidisk

In general, the Beurling theorem does not hold for an invariant subspace in the Hardy space over the bidisk. In 1991, Nakazi posed a conjecture that the Beurling theorem holds for a singly generated invariant subspace. In this paper, a relation between a singly generated invariant subspace and a weighted Hardy space over the bidisk is studied. It is showed that there exists a weighted Hardy space over the bidisk which has a non-cyclic reproducing kernel. Also a counterexample for Nakazi's conjecture is given.

A noncommutative Beurling theorem with respect to unitarily invariant norms

A noncommutative Beurling theorem with respect to unitarily invariant norms

Erratum to the article ``Beurling's theorem for nilpotent Lie groups'' Osaka J. Math. 48 (2011), 127--147

Erratum to the article ``Beurling's theorem for nilpotent Lie groups'' Osaka J. Math. 48 (2011), 127--147

Extensions of Beurling's prime number theorem

Kahane provided a proof of a conjecture of Bateman and Diamond on Beurling generalized primes. The conjecture says that the L2-condition [Formula: see text] implies the PNT, i.e. π(x) ~ x/ log x. Recently, Schlage-Puchta and Vindas showed that Beurling's condition in the form of a Cesàro mean [Formula: see text] with γ > 3/2 implies the PNT as well. Both are extensions of the Beurling PNT. We now show that the condition [Formula: see text] implies the PNT and that [Formula: see text] with k ≥ 0 implies the PNT also, where [Formula: see text]These results cover the preceding forms and extend further the Beurling PNT. As part of our argument, we extend first the theorems on Chebyshev bounds with correspondingly weaker conditions. These bounds play an important role in later proofs.

Variations on a theorem of Beurling

AbstractWe consider functions satisfying the subcritical Beurling's condition, viz.,

A constructive proof of Beurling-Lax theorem

This paper deals with an alternative proof of Beurling-Lax theorem by adopting a constructive approach instead of the isomorphism technique which was used in the original proof.

On Invariant Subspaces in Weighted Hardy Spaces

We consider the description of translation invariant subspaces of a weighted Hardy space in the half plane. The obtained result includes the Beurling–Lax theorem for the Hardy space as a special case. We discuss the problem of generalization of the definition of inner function.