5ECM Minisymposium

Spectral problems and Hilbert spaces of entire functions

Organizers: Joaquim Bruna, Hċkan 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.