649 research outputs found
Schur analysis in the quaternionic setting: The fueter regular and the slice regular case
No full textThis chapter is a survey on recent developments in quaternionic Schur analysis. The first part is based on functions which are slice hyperholomorphic in the unit ball of the quaternions, and have modulus bounded by 1. These functions, which by analogy to the complex case are called Schur multipliers, are shown to be (as in the complex case) the source of a wide range of problems of general interest. They also suggest new problems in quaternionic operator theory, especially in the setting of indefinite inner product spaces. This chapter gives an overview on rational functions and their realizations, on the Hardy space of the unit ball, on the half-space of quaternions with positive real part, and on Schur multipliers, also discussing related results. For the purpose of comparison this chapter presents also another approach to Schur analysis in the quaternionic setting, in the framework of Fueter series. To ease the presentation most of the chapter is written for the scalar case, but the reader should be aware that the appropriate setting is often that of vector-valued functions
A Hörmander–Fock space
No full textIn a recent paper, we used a basic decomposition property of poly analytic functions of order 2 in one complex variable to characterize solutions of the classical partial derivative(-)-problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hormander-Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by eta= (eta(n))(n= 0) leading to a special entire function E(z) that is used to express the kernel function of the Hormander-Fock space. We present also an example of a special function belonging to the class Mittag-Leffler (ML) introduced recently by Alpay et al. and apply a Bochner-Minlos type theorem to this function, thus motivating further connections with the theory of stochastic processes
Fock and Hardy spaces: Clifford Appell case
No full textIn this paper, we study a specific system of Clifford-Appell polynomials and, in particular, their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows us to obtain various function spaces by specifying a suitable sequence of real numbers. We focus on the Fock and Hardy cases in this setting, and we study the action of the Fueter mapping and its range
On the global operator and Fueter mapping theorem for slice polyanalytic functions
No full textIn this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball
Slice hyperholomorphic functional calculi
No full textThis paper is a survey on some functional calculi constructed with slice hyperholomorphic functions, mainly on the S-functional calculus. This is a functional calculus, defined for n-tuples of not necessarily commuting operators, and it is based on the recent theory of slice hyperholomorphic functions. Its version for commuting operators, called the SC-functional calculus, and its quaternionic version, called the quaternionic functional calculus, are presented, as well as the so-called F-functional calculus, based on the Fueter-Sce mapping theorem in integral form. Since the theory of hyperholomorphic functions is quite recent, the paper contains the main results of this function theory that are necessary to introduce these calculi, as well as the Fueter-Sce mapping theorem in integral form
Herglotz functions: The Fueter variables case
No full textIn this paper we provide realizations of Herglotz functions in the quaternionic case where hyperholomorphic functions are meant in the sense of Fueter. We consider the counterparts of the Arveson space and of the Herglotz–Agler functions by introducing suitable positive kernels. Crucial tools are the so-called Fueter variables and the CK-product. Our realization results also show that positivity implies analyticity
Realizations of Holomorphic and Slice Hyperholomorphic Functions: The Krein Space Case
No full textIn this paper we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space and may open new avenues of research in hypercomplex analysis
The Fock Space as a De Branges–Rovnyak Space
No full textWe show that de Branges–Rovnyak spaces include as special cases a number of spaces, such as the Hardy space, the Fock space, the Hardy–Sobolev space and the Dirichlet space. We present a general framework in which all these spaces can be obtained by specializing a sequence that appears in the construction. We show how to exploit this approach to solve interpolation problems in the Fock space
Superoscillations and Analytic Extension in Schur Analysis
No full textWe give applications of the theory of superoscillations to various questions, namely extension of positive definite functions, interpolation of polynomials and also of Rfunctions; we also discuss possible applications to signal theory and prediction theory of stationary stochastic processes. In all cases, we give a constructive procedure, by way of a limiting process, to get the required results
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