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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.
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The Nevanlinna parametrization establishes a bijection between the class of all measures having a prescribed set of moments and the class of Pick functions. The fact that all measures constructed through the Nevanlinna parametrization have identical moments follows from the theory of orthogonal polynomials and continued fractions. In this paper we explore the opposite direction: we construct a set of measures and we show that they all have identical moments, and then we establish a Nevanlinna-type parametrization for this set of measures. Our construction does not require the theory of orthogonal polynomials and it exposes the analytic structure behind the Nevanlinna parametrization.
In this note, we obtain a full characterization of radial Carleson measures for the Hilbert-Hardy space on tube domains over symmetric cones. For large derivatives, we also obtain a full characterization of the measures for which the corresponding embedding operator is continuous. Restricting to the case of light cones of dimension three, we prove that by freezing one or two variables, the problem of embedding derivatives of the Hilbert-Hardy space into Lebesgue spaces reduces to the characterization of Carleson measures for Hilbert-Bergman spaces of the upper-half plane or the product of two upper-half planes.
We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichlet space functions. As a by-product of the methods of proof, the results also hold in the vector-valued setting.
In this note, we frst consider boundedness properties of a family of operators generalizing the Hilbert operator in the upper triangle case. In the diagonal case, we give the exact norm of these operators under some restrictions on the parameters. We secondly consider boundedness properties of a family of positive Bergman-type operators of the upper-half plane. We give necessary and sufficient conditions on the parameters under which these operators are bounded in the upper triangle case.
We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions.
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