Function Spaces, Theory and Applications

The focus program on Analytic Function Spaces and their Applications took place at Fields Institute from July 1st to December 31st, 2021. Hilbert spaces of analytic functions form one of the pillars of complex analysis. These spaces have a rich structure and for more than a century have been studied by many prominent mathematicians. They also have several essential applications in other fields of mathematics and engineering, e.g., robust control engineering, signal and image processing, and theory of communication. The most important Hilbert space of analytic functions is the Hardy class H2. However, its close cousins, e.g. the Bergman space A2, the Dirichlet space D, the model subspaces Kt, and the de Branges-Rovnyak spaces H(b), have also been the center of attention in the past two decades. Studying the Hilbert spaces of analytic functions and the operators acting on them, as well as their applications in other parts of mathematics or engineering were the main subjects of this program. During the program, the world leading experts on function spaces gathered and discussed the new achievements and future venues of research on analytic function spaces, their operators, and their applications in other domains.

With more than 250 hours of lectures by prominent mathematicians, a wide variety of topics were covered. More explicitly, there were mini-courses and workshops on Hardy Spaces, Dirichlet Spaces, Bergman Spaces, Model Spaces, Interpolation and Sampling, Riesz Bases, Frames and Signal Processing, Bounded Mean Oscillation, de Branges-Rovnyak Spaces, Operators on Function Spaces, Truncated Toeplitz Operators, Blaschke Products and Inner Functions, Discrete and Continuous Semigroups of Composition Operators, The Corona Problem, Non-commutative Function Theory, Drury-Arveson Space, and Convergence of Scattering Data and Non-linear Fourier Transform. At the end of each week, there was a high profile colloquium talk on the current topic. The program also contained two semester-long advanced courses on Schramm Loewner Evolution and Lattice Models and Reproducing Kernel Hilbert Space of Analytic Functions. The current volume features a more detailed version of some of the talks presented during the program.

Consolidates over 250 hours of lectures by prominent mathematicians Explains operators acting on Hilbert spaces, as well as their applications in other parts of mathematics and engineering Broadens reader understanding of Hilbert spaces of analytic functions

Auteur
Ilia Binder is a professor of Mathematics at the University of Toronto. His research interests are Geometric Function Theory, Complex Analysis and Potential Theory, Complexity and Computability in Analysis and Dynamical Systems, Complex Dynamics, Stochastic Löwner Evolution, Multifractal Analysis, Quasi-periodic and random Shroedinger and Jacobi operators. He is also the Chair of the Mathematical and Computational Sciences Department at the University of Toronto Mississauga.

Damir Kinzebulatov is an associate professor at Laval University. He is working in the field of singular stochastic dynamics and regularity theory of PDEs with rough coefficients, which includes heat kernel bounds for local and non-local Kolmogorov operators, gradient estimates, unique continuation for eigenfunctions of Schroedinger operators (using methods of Nash, De Giorgi, Carleman). He uses harmonic-analytic and operator-theoretic methods in concrete settings.

Javad Mashreghi is a mathematician and author working in fields of function space theory, functional analysis and complex analysis. He is a professeur titulaire at Laval University and was recently the 35th President of the Canadian Mathematical Society. He is immensely involved in various aspects of North America's mathematical community, having served on numerous editorial, administrative and selection committees. He is the editor-in-chief of the Canadian Mathematical Bulletin and Concrete Operators, and the Analysis Section Coordinating Editor of the Proceedings of the American Mathematical Society. Among his awards and distinctions, there are the IEEE Prize Paper Award 2021, the Fellow of the Canadian Mathematical Society 2019, the Khwarizmi International Award, Research Prize of IROST, 2009, and the G. de B. Robinson Award 2004. As for publications, he has more than 120 articles, 12 books published by Cambridge, Oxford, AMS and Springer, and 11 conference proceedings.

Contenu

Absolute continuity in higher dimensions.- An indefinite analog of Sarason's generalized interpolation theorem.- An Operator theoretical approach of some inverse problems.- Applications of the automatic additivity of positive homogenous order isomorphisms between positive definite cones in C*-algebras.- Direct and Inverse Spectral Theorems for a Class of Canonical Systems with two Singular Endpoints.- Nevanlinna domains and uniform approximation by polyanalytic polynomial modules.- On meromorphic inner functions in the upper half-plane.- On the norm of the Hilbert matrix.- Radial limits of functions holomorphic in C or the polydisc.- Recent developments in the interplay between function theory and operator theory for block Toeplitz, Hankel, and model operators.- Sarason's Ha-plitz product problem.- Sub-Hardy Hilbert spaces in the non-commutative unit row-ball.- The relationship of the Gaussian curvature with the curvature of a Cowen-Douglas operator.- Weighted Polynomial Approximation on the Cubes of the non-zero Integers.- Index.