Reproducing kernels in Analysis and beyond – REPKA

Our project consists of several interrelated tasks dealing with topical problems in modern complex analysis, operator theory and their important applications to other fields of mathematics including approximation theory, probability, and control theory. The project is centered around the notion of the so-called reproducing kernel of a Hilbert space of holomorphic functions. Reproducing kernels are very powerful objects playing an important role in numerous domains such as determinantal point processes, signal theory, Sturm-Liouville and Schrödinger equations.

The advent of reproducing kernels goes back to the founding works of Nevanlinna, Pick, and Schur on exact constrained interpolation. Since then, reproducing kernels and pertaining techniques repeatedly made evidence of their central rôle and strength in the study of substantial properties of different spaces of holomorphic functions as demonstrated by breakthrough results on interpolation, sampling, uniqueness, invariant subspaces by Carleson, Seip, Aleman-Richter-Sundberg, etc. For recent interesting applications of Bergman, Fock and Paley-Wiener spaces techniques see the work by Gutman-Tsukamoto on embedding minimal dynamical systems (using sampling in the Paley-Wiener space) by Bufetov-Qiu-Shamov on determinantal point processes (dealing with complete minimal systems of reproducing kernels in the Paley-Wiener and in the Fock space), and by Belov-Borichev solving an old problem of Newman-Shapiro (on the multiplication operators in the Fock space).

Reproducing kernels, in a multitude of their facets and applications, form the unifying thread of the current project. The originality of this project is to bring together renowned experts from different fields: operator theory specialists, complex and harmonic analysists, experts in determinantal point processes and in several complex variables and two colleagues from INRIA, who will work together in the broadest possible configuration.

Geometric properties of families of reproducing kernels in Hilbert spaces of holomorphic functions like completeness, minimality, being a Riesz basis, etc. are intimately related to properties like interpolation, sampling, uniqueness in spaces of holomorphic functions. We are mainly interested here in Bergman, Fock, Paley-Wiener and, more generally, de Branges spaces. These properties will also be discussed in a random setting, which, for applications, can be a convenient situation. In our study of determinant point processes induced by the reproducing kernels of weighted Bergman space
we are specially interested in the reconstruction of a function by the values on the random point set and in describing the conditional measures with respect to fixing the configuration in a part of the phase space.

We will also investigate reproducing kernels in non-holomorphic setting, in particular in generalized Hardy classes arising in the context of conductivity partial differential equations in domains of dimension 2. They have far-reaching applications to problems in plasma shaping and in thermonuclear physics. In a first step we seek to characterize reproducing kernels in this setting (depending on smoothness properties of the conductivity), and then try to obtain geometric properties of families of such kernels.

Properties of interpolation, sampling and uniqueness are also considered in a more general Banach setting. We are in particular interested here in spaces of holomorphic functions arising in mathematical physics, like in Schrödinger, Dirac, Pauli operators and Jacobi matrices. In many problems on non-selfadjoint perturbations of these operators, the problem of the discrete spectrum can be translated into a Blaschke type condition which yields Lieb-Thirring type inequalities. Reproducing kernels are also related to Hankel and Toeplitz operators (Berezin transform). We will study the spectral theory of unbounded Toeplitz operators on Bergman spaces and that of Hankel operators on Fock spaces.