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Irene Sabadini

Publications and source records attributed to Irene Sabadini.

At least 73 records · Page 4Linked to original sources
arXiv2017

On the Gauss-Lucas theorem in the quaternionic setting

In theory of one complex variable, Gauss-Lucas Theorem states that the critical points of a non constant polynomial belong to the convex hull of the set of zeros of the polynomial. The exact analogue of this result cannot hold, in general, in the quaternionic case; instead, the critical points of a non constant polynomial belong to the convex hull of the set of zeros of the so-called symmetrization of the given polynomial. An incomplete proof of this statement was given in [8]. In this paper we present a different but complete proof of this theorem and we discuss a consequence.

math.CV↗
arXiv2017

Segal-Bargmann-Fock modules of monogenic functions

In this paper we introduce the classical Segal-Bargmann transform starting from the basis of Hermite polynomials and extend it to Clifford algebra-valued functions. Then we apply the results to monogenic functions and prove that the Segal-Bargmann kernel corresponds to the kernel of the Fourier-Borel transform for monogenic functionals. This kernel is also the reproducing kernel for the monogenic Bargmann module.

math.CV↗
arXiv2017

Perturbation of normal quaternionic operators

The theory of quaternionic operators has applications in several different fields such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of spectrum. In fact, in quaternionic operator theory the classical notion of resolvent operator and the one of spectrum need to be replaced by the two $S$-resolvent operators and the $S$-spectrum. This is a consequence of the non-commutativity of the quaternionic setting. Indeed, the $S$-spectrum of a quaternionic linear operator $T$ is given by the non invertibility of a second order operator. This presents new challenges which makes our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of $S$-spectrum and of slice hyperholomorphicity of the $S$-resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator.

math.FA↗
arXiv2017

Carleson measures for Hardy and Bergman spaces in the quaternionic unit ball

We study a characterization of slice Carleson measures and of Carleson measures for the both the Hardy spaces $H^p(\mathbb B)$ and the Bergman spaces $\mathcal A^p(\mathbb B)$ of the quaternionic unit ball $\mathbb B$. In the case of Bergman spaces, the characterization is done in terms of the axially symmetric completion of a pseudohyperbolic disc in a complex plane. We also show that a characterization in terms of pseudohyperbolic balls is not possible.

math.CV↗
arXiv2016

On the Bargmann-Radon transform in the monogenic setting

In this paper we introduce and study a Bargmann-Radon transform on the real monogenic Bargmann module. This transform is defined as the projection of the real Bargmann module on the closed submodule of monogenic functions spanned by the monogenic plane waves. We prove that this projection can be written in integral form in terms the so-called Bargmann-Radon kernel. Moreover, we have a characterization formula for the Bargmann-Radon transform of a function in the real Bargmann module in terms of its complex extension and then its restriction to the nullcone in $\mathbb C^m$. We also show that the formula holds for the Szegő-Radon transform that we introduced in [4] (see list of references). Finally, we define the dual transform and we provide an inversion formula.

math.CV↗
arXiv2016

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 Carathéodory theorems in this setting.

math.CV↗
arXiv2016

Fueter's theorem for monogenic functions in biaxial symmetric domains

In this paper we generalize the result on Fueter's theorem from [10] by Eelbode et al. to the case of monogenic functions in biaxially symmetric domains. To obtain this result, Eelbode et al. used representation theory methods but their result also follows from a direct calculus we established in our paper [21]. In this paper we first generalize [21] to the biaxial case and derive the main result from that.

math.CV↗
arXiv2016

On two-sided monogenic functions of axial type

In this paper we study two-sided (left and right) axially symmetric solutions of a generalized Cauchy-Riemann operator. We present three methods to obtain special solutions: via the Cauchy-Kowalevski extension theorem, via plane wave integrals and Funk-Hecke's formula and via primitivation. Each of these methods is effective enough to generate all the polynomial solutions.

math.CV↗
arXiv2016

Adaptative decomposition: the case of the Drury-Arveson space

The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space $\mathbf H_2$ of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and products have counterparts in the unit ball of $\mathbb C^N$, and this fact allows us to extend in the present paper the maximum selection principle to the case of functions in the Drury-Arveson space of functions analytic in the unit ball of $\mathbb C^N$. This will give rise to an algorithm which is a variation in this higher dimensional case of the greedy algorithm. We also introduce infinite Blaschke products in this setting and study their convergence.

math.CV↗
arXiv2015

Beurling-Lax type theorems in the complex and quaternionic setting: the half-space case

We give a generalization of the Beurling-Lax theorem both in the complex and quaternionic settings. We consider in the first case functions meromorphic in the right complex half-plane, and functions slice hypermeromorphic in the right quaternionic half-space in the second case. In both settings we also discuss a unified framework, which includes both the disk and the half-plane for the complex case and the open unit ball and the half-space in the quaternionic setting.

math.FA↗
arXiv2015

Wiener algebra for the quaternions

We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-Lévy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener-Hopf operators.

math.CV↗
arXiv2014

On Some Geometric Properties of Slice Regular Functions of a Quaternion Variable

The goal of this paper is to introduce and study some geometric properties of slice regular functions of quaternion variable like univalence, subordination, starlikeness, convexity and spirallikeness in the unit ball. We prove a number of results, among which an Area-type Theorem, Rogosinski inequality, and a Bieberbach-de Branges Theorem for a subclass of slice regular functions. We also discuss some geometric and algebraic interpretations of our results in terms of maps from $\mathbb R^4$ to itself. As a tool for subordination we define a suitable notion of composition of slice regular functions which is of independent interest.

math.CV↗
arXiv2014

Generalized quaternionic Schur functions in the ball and half-space and Krein-Langer factorization

In this paper we prove a new version of Krein-Langer factorization theorem in the slice hyperholomorphic setting which is more general than the one proved in [D. Alpay, F. Colombo, I. Sabadini, Krein-Langer factorization and related topics in the slice hyperholomorphic setting, J. Geom. Anal., 24 (2014), 843--872]. We treat both the case of functions with $κ$ negative squares defined on subsets of the quaternionic unit ball or on subsets of the half space of quaternions with positive real part. A crucial tool in the proof of our results is the Schauder-Tychonoff theorem and an invariant subspace theorem for contractions in a Pontryagin space.

math.CV↗
arXiv2014

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.

math.CV↗
arXiv2014

Approximation by polynomials on quaternionic compact sets

In this paper we obtain several extensions to the quaternionic setting of some results concerning the approximation by polynomials of functions continuous on a compact set and holomorphic in its interior. The results include approximation on compact starlike sets and compact axially symmetric sets. The cases of some concrete particular sets are described in details, including quantitative estimates too.

math.CV↗
arXiv2014

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.

math.CV↗
arXiv2014

Further properties of the Bergman spaces of slice regular functions

In this paper we continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paper we mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitable weights are taken into account. Finally, we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane.

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