BMO‐ and VMO‐spaces of slice hyperholomorphic functions

In this paper we continue the study of important Banach spaces of slice hyperholomorphic functions on the quaternionic unit ball by investigating the BMO‐ and VMO‐spaces of slice hyperholomorphic functions. We discuss in particular conformal invariance and a refined characterization of these spaces in terms of Carleson measures. Finally we show the relations with the Bloch and Dirichlet space and the duality relation with the Hardy space . The importance of these spaces in the classical theory is well known. It is therefore worthwhile to study their slice hyperholomorphic counterparts, in particular because slice hyperholomorphic functions were found to have several applications in operator theory and Schur analysis.     

Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting

In this paper we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allow to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels and positive definite functions in this setting and we show how they can be obtained using the extension operator and the slice hyperholomorphic product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges–Rovnyak space.     

The Cholewinski‐Fock space in the slice hyperholomorphic setting

Inspired from the Cholewinski approach, we investigate a family of Fock spaces in the quaternionic slice hyperholomorphic setting as well as some associated quaternionic linear operators. In a particular case, we reobtain the slice hyperholomorphic Fock space on quaternions.     

Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions. These 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 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 Carathéodory functions.     

The Slice Hyperholomorphic Bergman Space on $$\varvec{\mathbb {B}_{R}}$$: Integral Representation and Asymptotic Behavior

The aim of the present paper is threefolds. Firstly, we complete the study of the weighted hyperholomorphic Bergman space of the second kind on the ball of radius R centred at the origin. The explicit expression of its Bergman kernel is given and can be written in terms of special hypergeometric functions of two non-commuting (quaternionic) variables. Secondly, we introduce and study some basic properties of an associated integral transform, the quaternionic analogue of the so-called second Bargmann transform for the holomorphic Bergman space. Finally, we establish the asymptotic behavior as R goes to infinity. We show in particular that the reproducing kernel of the weighted slice hyperholomorphic Bergman space gives rise to its analogue for the slice hyperholomorphic Bargamann–Fock space.     

Krein–Langer Factorization and Related Topics in the Slice Hyperholomorphic Setting

We study various aspects of Schur analysis in the slice hyperholomorphic setting. We present two sets of results: first, we give new results on the functional calculus for slice hyperholomorphic functions. In particular, we introduce and study some properties of the Riesz projectors. Then we prove a Beurling–Lax type theorem, the so-called structure theorem. A crucial fact which allows to prove our results is the fact that the right spectrum of a quaternionic linear operator and the point S-spectrum coincide. Finally, we study the Krein–Langer factorization for slice hyperholomorphic generalized Schur functions. Both the Beurling–Lax type theorem and the Krein–Langer factorization are far-reaching results which have not been proved in the quaternionic setting using notions of hyperholomorphy other than slice hyperholomorphy.     

Interpolation Problems for Certain Classes of Slice Hyperholomorphic Functions

A general interpolation problem (which includes as particular cases the Nevanlinna–Pick and Carathéodory–Fejér interpolation problems) is considered in two classes of slice hyperholomorphic functions of the unit ball of the quaternions. In the Hardy space of the unit ball we present a Beurling–Lax type parametrization of all solutions, and the formula for the minimal norm solution. In the class of functions slice hyperholomorphic in the unit ball and bounded by one in modulus there (that is, in the class of Schur functions in the present framework) we present a necessary and sufficient condition for the problem to have a solution, and describe the set of all solutions in the indeterminate case.     

On the Structure of the Taylor Series in Clifford and Quaternionic Analysis

The Fueter variables form a basis of the space of (quaternionic or Cliffordian) hyperholomorphic homogeneous polynomials of degree one, and their symmetrized products give the respective bases of spaces of hyperholomorphic homogeneous polynomials for any degree k. In the present paper we introduce new bases, i.e., new types of hyperholomorphic variables which lead to the Taylor-type series expansions reflecting the structure of the set of all (quaternionic or Cliffordian algebra-valued) hyperholomorphic functions.     

Schatten class and Berezin transform of quaternionic linear operators

In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S‐spectrum of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S‐functional calculus. Copyright © 2016 John Wiley & Sons, Ltd.     

On discrete analytic functions: Products, rational functions and reproducing kernels

We introduce a family of discrete analytic functions, called expandable discrete analytic functions, which includes discrete analytic polynomials, and define two products in this family. The first one is defined in a way similar to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us to define rational discrete analytic functions. To define the second product we need a new space of entire functions which is contractively included in the Fock space. We study in this space some counterparts of Schur analysis.     

Espaces de Branges Rovnyak et fonctions de Schur : le cas hyper-analytique

We extend to the hyperholomorphic case the notion of Schur functions and the corresponding realization theory. We introduce the notion of characteristic operator function for coisometric colligations between Hilbert spaces of hyperholomorphic functions. We show that every Schur function is the characteristic operator function of a coisometric colligation and vice-versa. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Lacunary Series in Weighted Hyperholomorphic B p,q (G) Spaces

The goal of this article is two-fold. First, we consider a class of hyperholomorphic functions, the so called B ^{p, q} (G) space in ?³. Then, we use the B ^{p, q} (G) space to characterize the hyperholomorphic α-Bloch space. Second, we obtain characterizations of the weighted hyperholomorphic B ^{p, q} (G)-functions by the coefficients of certain lacunary series expansions in Clifford Analysis.     

Gleason's problem and tangential homogeneous interpolation for hyperholomorphic quaternionic functions

We study a version of Gleason's problem in the setting of functions of class C¹ in the unit ball of ?². We use the setting of hyperholomorphic functions to define and solve the problem. Finally, we briefly discuss a tangential interpolation problem for hyperholomorphic functions.     

关于ω-强凸空间的一点注记

本文研究了关于ω-强凸空间和ω-强光滑空间的问题.利用Banach理论的方法,证明了ω-强凸空间和ω-强光滑空间是一对对偶概念,并讨论了ω-强光滑性与其它光滑性之间的关系,用切片统一刻画了ω-强凸空间与ω-强光滑空间的特征,完善了ω-强凸空间及其对偶空间的研究.     

Applications of {\Phi} -Operators to Hypercomplex Geometry

The main aim of this paper is to study relations between F{\Phi} -operators, hyperholomorphic functions and the theory of lifts in tensor bundles. Model of lifts of affinor fields was found by using of F{\Phi} -operators. After that, hyperholomorphic properties of pure cross-sections were investigated by the help of this model.     

Quaternionic spherical wave functions

The scalar spherical wave functions (SWFs) are solutions to the scalar Helmholtz equation obtained by the method of separation of variables in spherical polar coordinates. These functions are complete and orthogonal over a sphere, and they can, therefore, be used as a set of basis functions in solving boundary value problems by spherical wave expansions. In this work, we show that there exists a theory of functions with quaternionic values and of three real variables, which is determined by the Moisil–Theodorescu‐type operator with quaternionic variable coefficients, and which is intimately related to the radial, angular and azimuthal wave equations. As a result, we explain the connections between the null solutions of these equations, on one hand, and the quaternionic hyperholomorphic and anti‐hyperholomorphic functions, on the other. We further introduce the quaternionic spherical wave functions (QSWFs), which refine and extend the SWFs. Each function is a linear combination of SWFs and products of ‐hyperholomorphic functions by regular spherical Bessel functions. We prove that the QSWFs are orthogonal in the unit ball with respect to a particular bilinear form. Also, we perform a detailed analysis of the related properties of QSWFs. We conclude the paper establishing analogues of the basic integral formulae of complex analysis such as Borel–Pompeiu's and Cauchy's, for this version of quaternionic function theory. As an application, we present some plot simulations that illustrate the results of this work. Copyright © 2016 John Wiley & Sons, Ltd.     

解析函数的Banach空间上之复合算子

本文研究了一类解析函数的Ｂａｎａｃｈ空间Ｘ上之复合算子，这类空间包含了Ｂｌｏｃｈ空间，并且可看作Ｂｅｒｇｍａｎ空间Ｌ１ａ（Ｄ）中具有原子分解的解析函数的对偶空间．我们刻划了这类空间上紧复合算子及Ｆｒｅｄｈｏｌｍ复合算子的特征，此外，还研究了具有闭值域的复合算子．     

On Geometric Aspects of Quaternionic and Octonionic Slice Regular Functions

The aim of this paper is twofold. On the one hand, we enrich from a geometrical point of view the theory of octonionic slice regular functions. We first prove a boundary Schwarz lemma for slice regular self-mappings of the open unit ball of the octonionic space. As applications, we obtain two Landau–Toeplitz type theorems for slice regular functions with respect to regular diameter and slice diameter, respectively, together with a Cauchy type estimate. Along with these results, we introduce some new and useful ideas, which also allow us to prove the minimum principle and one version of the open mapping theorem. On the other hand, we adopt a completely new approach to strengthen a version of boundary Schwarz lemma first proved in Ren and Wang (Trans Am Math Soc 369:861–885, 2017) for quaternionic slice regular functions. Our quaternionic boundary Schwarz lemma with optimal estimate improves considerably a well-known Osserman type estimate and provides additionally all the extremal functions.     

TWO POROSITY RESULTS IN MONOTONIC ANALYSIS

ABSTRACT

In this work we consider spaces of increasing functions defined on a subset of an ordered normed space. We equip each of these spaces with a natural metric and show that the complement of the subset of all strictly increasing functions is σ-porous. We also discuss some properties of normal sets and strictly normal sets.     

A Gleason–Kahane–Żelazko theorem for the Dirichlet space

We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of weighted composition operators on the Dirichlet space as being exactly those linear maps that send nowhere-vanishing functions to nowhere-vanishing functions.We also investigate possible extensions to weighted Dirichlet spaces with superharmonic weights. As part of our investigation, we are led to determine which of these spaces contain functions that map the unit disk onto the whole complex plane.