Pólya sequences, Toeplitz kernels and gap theorems

Abstract

A separated sequence Λ on the real line is called a Pólya sequence if any entire function of zero exponential type bounded on Λ is constant. In this paper we solve the problem by Pólya and Levinson that asks for a description of Pólya sets. We also show that the Pólya-Levinson problem is equivalent to a version of the so-called Beurling gap problem on Fourier transforms of measures. The solution is obtained via a recently developed approach based on the use of Toeplitz kernels and de Branges spaces of entire functions.1.2. Background. We use the standard notation N + (C + ) to denote the Smirnov-Nevanlinna class in the upper half-plane C + = {z|ℑz > 0} consisting of analytic functions f (z) that can be represented as a ratio g(z)/h(z) of two bounded analytic functions with h(z) being outer. Each function in N + (C + ) has non-tangential

A problem on completeness of exponentials

“…As was shown in [27], for any finite measure µ on R, G 1 µ , as defined in the previous section, depends only on its support:…”

“…By duality, for 1 < p ∞, G p µ can still be defined as the infimum of a such that E a is complete in L q (µ), 1 p + 1 q = 1. The cases p = 2 were considered in several papers; see, for instance, articles by Koosis [16] or Levin [20] for the case p = ∞ or [27] for p = 1.…”

Restricted interpolation by meromorphic inner functions

“…A more general result, giving a formula for the maximal size of the gap in the Fourier spectrum for all measures supported on a fixed set was recently found in [25] (see also [27]), however these results have much more complicated proofs. In the case of a separated sequence ƒ it was shown by M. Mitkovski and the first author in [22] (see also [27]), that a necessary and sufficient condition for ƒ to support a measure with a spectral gap is that its interior Beurling-Malliavin density It is, therefore, natural to ask which sequences ƒ have the property as demanded by de Branges. In [30], the second author addresses this question and provides answers for a large class of sequences.…”

“…It is known (see [27], [22]) that the exterior Beurling-Malliavin density of a discrete sequence ƒ is equivalent with the following definition. With these definitions, we can state the precise result here.…”

“…The class of MIFs serves as a natural object in the so called Toeplitz approach to the Uncertainty Principle, see ([19], [20], [27]). It was used to obtain an extension of the Beurling-Malliavin theory ( [19], [20]) and to study classical problems of Fourier analysis ( [22], [23], [25], [26]). Some function theoretic questions arising from such applications were treated in [30].…”