Spectral problems and Hilbert spaces of entire functions

Organizers: Joaquim Bruna, Håkan Hedenmalm, Kristian Seip, and Mikhail Sodin

Speakers

Nikolai Makarov, California Institute of Technology, USA
Jean-François Burnol, University of Lille 1, France
Aharon Atzmon, Tel Aviv University, Israel
Alexander Ulanovskii, University of Stavanger, Norway
Alexander Borichev, University of Provence, France

Background

Hilbert spaces of entire functions were first studied by Paley and Wiener in their classical treatise from 1934. Around 1960, fundamental results on multipliers and completeness were obtained by Beurling and Malliavin in two of the deepest works in complex analysis in the twentieth century. Around the same time, a beautiful and unifying theory of Hilbert spaces of entire functions was developed by de Branges. Since then, such spaces are ubiquitous in harmonic analysis, the theory of entire functions, and in spectral theory of Sturm-Liouville operators and Krein strings.

During the last 10 years, Hilbert spaces of entire functions have appeared prominently in several seemingly unrelated developments:

The discovery of new links between inverse spectral problems for Krein strings, completeness of exponentials, de Branges spaces, and the theory of model subspaces K_θ of the classical Hardy space H² (Makarov and Poltoratski).
Relations between de Branges spaces and Dirichlet-Riemann L-functions (Burnol, Lagarias).
Description of the exponential frames on an interval (Ortega-Cerdà and Seip) and a related study of sampling and interpolation in de Branges spaces (Lyubarskii and Seip). Stable reconstruction of band-limited signals with given disconnected spectrum (Olevskii and Ulanovskii).
New links between Hilbert spaces of entire functions beyond the Cartwright class, the uncertainty principle, and translation invariant subspaces of families of weighted spaces (Atzmon).
The study of the indeterminate case of the Hamburger moment problem and of the Bernstein weighted polynomial approximation problem (Borichev and Sodin).

An important goal for the minisymposium is to provide crash introductions to some of these achievements for a wide audience of analysts. Another purpose is to promote interaction between researchers from different areas sharing an interest in Hilbert spaces of entire functions and their applications. The speakers represent both the wide scope of the topic and important new results. The minisymposium will be an excellent opportunity to shed light on the results of these researchers and to see their work in a broader perspective.

Schedule

1:25 - 2:10 p.m. Makarov
2:20 - 2:50 p.m. Burnol
3:10 - 3:40 p.m. Atzmon
3:45 - 4:15 p.m. Ulanovskii
4:20 - 4:50 p.m. Borichev

Titles and abstracts of talks

Nikolai Marakov: Linear complex analysis and de Branges spaces: Abstract: De Branges spaces of entire functions are closely related to the spectral theory of selfadjoint differential operators and the corresponding complex Fourier transforms. This is a formidable structure appearing in several classical problems of one-dimensional linear complex analysis. The techniques of Toeplitz operators and Aleksandrov-Clark measures provide effective tools in the study of de Branges spaces. I will review applications to weighted approximation, problems of completeness and minimality, and some aspects of the inverse spectral problem. The talk will be based on a joint work with Alexei Poltoratski.
Aharon Atzmon: Weighted Hardy spaces and the uncertainty principle for Fourier transforms: Abstract: We associate with certain weight functions on the real line some weighted Hardy spaces and some Banach spaces of entire functions, which can be identified with closed subspaces of the corresponding weighted Lebesgue spaces on the real line. We show that these spaces can be characterized by the rate of decay at infinity of the Fourier transforms of their elements and thereby obtain in a general setting extensions of the classical Paley-Wiener theorems and sharp forms of the uncertainty principle for Fourier transforms which imply most of the known results in this area.
Alexander Borichev: Approximation on the line: weight's perturbations: Abstract: We are going to discuss several results on the stability of weighted (polynomial) approximation under perturbations of the weight function obtained recently in a joint work with M.Sodin. Time permitting, we'll talk about an application to the inverse spectral problem.
Jean-Francois Burnol: The Fourier transform as a spectral problem: Abstract: The Fourier transform is mostly used as a tool. Number Theory demands to consider it also as an object of study. Aspects of such a study will be reported upon, where the Fourier transform receives an incarnation as a scattering. Some Hilbert spaces of entire functions arise in this study and are put in a new spectral light, which reveals their connection to a wider picture, provides new results, and allows an easier access to old ones which had for many decades remained cloaked in an hostile opacity.
Alexander Ulanovskii: Universal sampling and interpolation of band-limited signals: Abstract: The talk is based on a joint research with Prof. Alexander Olevskii. We ask if there exist discrete "universal" sets L of given finite density such that every signal f(t) with bounded spectrum of small measure can be recovered from the samples f(l), l in L. We prove that uniqueness in this problem can be achieved in general situation. On the other hand, for stable reconstruction it is crucial whether the spectrum is compact or dense.