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In this paper we obtain several extensions to the quaternionic setting of someresultsconcerningtheapproximation bypolynomials of functionscontinuous onacompact set and holomorphic in its interior. The results include approximationon compact starlike sets and compact axially symmetric sets. The cases of someconcrete particular sets are described in details, including quantitative estimatestoo. AMS 2010 Mathematics Subject Classification: Primary 30G35; Secondary 30E10,41A25.Keywords and phrases: Mergelyan’s theorem, quaternions, Riemann mapping, axiallysymmetric sets, approximation by polynomials, convolution operators, Cassini pseudo-metric, Cassini cell, order of approximation, slice regular functions. 1 Introduction It is well-known the fact that the Mergelyan’s approximation theorem is the ultimatedevelopment and generalization of the Weierstrass approximation theorem and Runge’stheorem in the complex plane. It can be stated as follows (see [13]):Theorem 1.1. Let K be a compact subset of the complex plane C such that C\ K isconnected. Then, every continuous function on K, f : K → C, which is holomorphic inthe interior of K, can be approximated uniformly on K by polynomials.Notice that all the known proofs of this result, based on the methods in complex analysisuse the Riemann mapping theorem.1

In a recent paper, we showed that the classical Bergman theory admits two possible formulations for the class of slice regular functions with quaternionic values. In the so called formulation of the first kind, we provide a Bergman kernel which is defined on and is a reproducing kernel. In the so called formulation of the second kind, we use the Representation Formula for slice regular functions to define a second Bergman kernel; this time the kernel is still defined on U, but the integral representation of f is based on an integral computed only on and the integral does not depend on , (here denotes the sphere unit of purely imaginary quaternions, and represents the complex plane with imaginary unit I). In this paper, we extend the second formulation of the Bergman theory to the case of slice monogenic functions and we focus our attention on the so-called Bergman–Sce transform. This integral transform is defined by using the Bergman kernel and the Sce mapping theorem and associates to every slice monogenic function f, an axially monogenic function . Copyright © 2011 John Wiley & Sons, Ltd.

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n-tuples of commuting operators (called ℱ-functional calculus) based on a new notion of spectrum, called ℱ-spectrum, for the n-tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ℱ-functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.

Recent advances in Computer Algebra have made it possible the study of algebraic analysis in an explicit and computational way. In this paper we show how these ideas have allowed the solution of a new class of problems in Clifford analysis and we describe the computational techniques that we have to develop to this purpose. Copyright © 2002 John Wiley & Sons, Ltd.

In this paper, we discuss the so-called Witt basis in a Clifford algebra and we axiomatically define an algebra of abstract Hermitian vector variables similar to the ‘radial algebra’. In this setting, we introduce some linear partial differential operators and we study their resolutions. Copyright © 2002 John Wiley & Sons, Ltd.