Pontryagin de Branges Rovnyak spaces of slice hyperholomorphic functions

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions which are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper we focus on the case of Hilbert spaces, and introduce in particular a version of the Hardy space. Then we define Blaschke factors and Blaschke products and we consider an interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, a theorem of Shmulyan on densely defined contractive linear relations. We then study realizations of generalized Schur functions and of generalized Carath'eodory functions

A Cauchy kernel for slice regular functions

In this paper we show how to construct a regular, non commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results

Entire slice regular functions

Entire functions in one complex variable are extremely relevant in several areas ranging from the study of convolution equations to special functions. An analog of entire functions in the quaternionic setting can be defined in the slice regular setting, a framework which includes polynomials and power series of the quaternionic variable. In the first chapters of this work we introduce and discuss the algebra and the analysis of slice regular functions. In addition to offering a self-contained introduction to the theory of slice-regular functions, these chapters also contain a few new results (for example we complete the discussion on lower bounds for slice regular functions initiated with the Ehrenpreis-Malgrange, by adding a brand new Cartan-type theorem). The core of the work is Chapter 5, where we study the growth of entire slice regular functions, and we show how such growth is related to the coefficients of the power series expansions that these functions have. It should be noted that the proofs we offer are not simple reconstructions of the holomorphic case. Indeed, the non-commutative setting creates a series of non-trivial problems. Also the counting of the zeros is not trivial because of the presence of spherical zeros which have infinite cardinality. We prove the analog of Jensen and Carath\'eodory theorems in this setting

The Radon transform between monogenic and generalized slice monogenic functions

In [J. Bures, R. Lavicka, V. Soucek, Elements of quaternionic analysis and Radon transform, Textos de Matematica 42, Departamento de Matematica, Universidade de Coimbra, 2009], the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on R^(n+1) using the Radon and dual Radon transforms. The main aim of this paper is to further develop this approach. In fact, the Radon transform for functions with values in the Clifford algebra R_n is mapping solutions of the generalized Cauchy-Riemann equation, i.e., monogenic functions, to a parametric family of holomorphic functions with values in R_n and, analogously, the dual Radon transform is mapping parametric families of holomorphic functions as above to monogenic functions. The parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so-called slice monogenic functions. An important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms

The Fock space in the slice hyperholomorphic setting

In this paper we introduce and study some basic properties of the Fock space (also known as Segal-Bargmann space) in the slice hyperholomorphic setting. We discuss both the case of slice regular functions over quaternions and also the case of slice monogenic functions with values in a Clifford algebra. In the specific setting of quaternions, we also introduce the full Fock space. This paper can be seen as the beginning of the study of infinite dimensional analysis in the quaternionic setting.Comment: to appear in "Hypercomplex Analysis: New Perspectives and Applications", Trends in Mathematics, Birkhauser, Basel, S. Bernstein et al. ed

Self-mappings of the quaternionic unit ball: multiplier properties, Schwarz-Pick inequality, and Nevanlinna--Pick interpolation problem

We study several aspects concerning slice regular functions mapping the quaternionic open unit ball into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space. In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case

Realizations of slice hyperholomorphic generalized contractive and positive functions

We introduce generalized Schur functions and generalized positive functions in setting of slice hyperholomorphic functions and study their realizations in terms of associated reproducing kernel Pontryagin spacesComment: Revised version,to appear in the Milan Journal of Mathematic