Dan Volok

Reproducing Kernel Spaces of Series of Fueter Polynomials

We study reproducing kernel spaces of power series of Fueter polynomials and their multipliers. In particular we prove a counterpart of Beurling-Lax theorem in the quaternionic Arveson space and we define and characterize counterparts of the Schur-Agler classes. We also address the notion of rationality in the hyperholomorphic setting.

Relative reproducing kernel Hilbert spaces

We introduce a reproducing kernel structure for Hilbert spaces of functions where differences of point evaluations are bounded. The associated reproducing kernels are characterized in terms of conditionally negative functions.

Fonctions rationnelles et théorie de la réalisation: le cas hyper-analytique

We define and study the ring of rational functions in the hyperholomorphic setting. We give a number of equivalent characterizations of rationality. The Cauchy–Kovalevskaya product plays an important role in the arguments. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Point Evaluation and Hardy Space on a Homogeneous Tree

Abstract.We consider stationary multiscale systems as defined by Basseville, Benveniste, Nikoukhah and Willsky. We show that there are deep analogies with the discrete time non stationary setting as developed by the first author, Dewilde and Dym. Following these analogies we define a point evaluation with values in a C*–algebra and the corresponding “Hardy space” in which Cauchy’s formula holds. This point evaluation is used to define in this context the counterpart of classical notions such as Blaschke factors.

Deformations of Fuchsian Systems of Linear Differential Equations and the Schlesinger System

We consider holomorphic deformations of Fuchsian systems parameterized by the pole loci. It is well known that, in the case when the residue matrices are non-resonant, such a deformation is isomonodromic if and only if the residue matrices satisfy the Schlesinger system with respect to the parameter. Without the non-resonance condition this result fails: there exist non-Schlesinger isomonodromic deformations. In the present article we introduce the class of the so-called isoprincipal deformations of Fuchsian systems. Every isoprincipal deformation is also an isomonodromic one. In general, the class of the isomonodromic deformations is much richer than the class of the isoprincipal deformations, but in the non-resonant case these classes coincide. We prove that a deformation is isoprincipal if and only if the residue matrices satisfy the Schlesinger system. This theorem holds in the general case, without any assumptions on the spectra of the residue matrices of the deformation. An explicit example illustrating isomonodromic deformations, which are neither isoprincipal nor meromorphic with respect to the parameter, is also given.

Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions I

In this second article of the series we study holomorphic families of generic rational matrix functions parameterized by the pole and zero loci. In particular, the isoprincipal deformations of generic rational matrix functions are proved to be isosemiresidual. The corresponding rational solutions of the Schlesinger system are constructed and the explicit expression for the related tau function is given. The main tool is the theory of joint system representations for rational matrix functions with prescribed pole and zero structures.

Évaluation ponctuelle et espace de Hardy : le cas multi-échelle

We define a point evaluation for transfer operators of multiscale causal dissipative systems. We associate to such a system a de Branges Rovnyak space, which serves as the state space of a coisometric realization.

A new realization of rational functions, with applications to linear combination interpolation, the Cuntz relations and kernel decompositions

We introduce the following linear combination interpolation problem (LCI), which in case of simple nodes reads as follows: given distinct numbers and complex numbers and , find all functions analytic in an open set (depending on ) containing the points such that To this end, we prove a representation theorem for such functions in terms of an associated polynomial . We give applications of this representation theorem to realization of rational functions and representations of positive definite kernels.

Pick Matrices for Schur Multipliers

It is well known that the classical Nevanlinna-Pick problem for holomorphic contractive functions in the open unit disk is solvable if and only if a matrix P with entries of the form