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

In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, and more generally to weighted Fock spaces.2000 Mathematics Subject Classification. 32A36, 32A, 47B05, 47B35.

In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator T are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying similar growth conditions. The main results extend the results of Xia and Zheng to the case of the Bergman space when 1 < p < ∞, and in the weighted Bargmann-Fock space setting, our results provide new, more general conditions that imply the work of Xia and Zheng via a more familiar approach that can also handle the 1 < p < ∞ case.2000 Mathematics Subject Classification. 32A36, 32A, 47B05, 47B35.

In this paper we characterize the compact operators on A p α (B n ) when 1 < p < ∞ and α > −1. The main result shows that an operator on A p α (B n ) is compact if and only if it belongs to the Toeplitz algebra and its Berezin transform vanishes on the boundary of the ball.

We study the general moment problem for measures on the real line, with polynomials replaced by more general spaces of entire functions. As a particular case, we describe measures that are uniquely determined by a restriction of their Fourier transform to a finite interval. We apply our results to prove an extension of a theorem by Eremenko and Novikov on the frequency of oscillations of measures with a spectral gap (high-pass signals) near infinity.