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About this book

This book presents the extensions to the quaternionic setting of some of the main approximation results in complex analysis. It also includes the main inequalities regarding the behavior of the derivatives of polynomials with quaternionic cofficients. With some few exceptions, all the material in this book belongs to recent research of the authors on the approximation of slice regular functions of a quaternionic variable.

The book is addressed to researchers in various areas of mathematical analysis, in particular hypercomplex analysis, and approximation theory. It is accessible to graduate students and suitable for graduate courses in the above framework.

Table of Contents

Frontmatter

Chapter 1. Preliminaries on Hypercomplex Analysis

Abstract
In this chapter we present some preliminary results which provide the background for the next chapters.
Sorin G. Gal, Irene Sabadini

Chapter 2. Approximation of Continuous Functions

Abstract
In this chapter we present general density results of Stone–Weierstrass type and of Carleman type in the space of continuous quaternionic-valued functions.
Sorin G. Gal, Irene Sabadini

Chapter 3. Approximation in Compact Balls by Bernstein and Convolution Type Operators

Abstract
In this chapter we present the quaternionic counterparts, in the slice regular case, of several celebrated results in complex approximation. In particular, we discuss approximation by Bernstein polynomials and by convolution operators in the case of compact balls and Cassini cells.
Sorin G. Gal, Irene Sabadini

Chapter 4. Approximation of Slice Regular Functions in Compact Sets

Abstract
In this chapter, we present the counterparts in the slice regular setting of some classical results in complex analysis, namely approximation results of Runge type, Mergelyan type and Arakelian type. Then, we study approximation by quaternionic Faber polynomials and by quaternionic polynomials in Bergman spaces.
Sorin G. Gal, Irene Sabadini

Chapter 5. Overconvergence, Equiconvergence and Universality Properties

Abstract
In this chapter we study in the quaternionic setting the overconvergence of Chebyshev and Legendre polynomials, the Walsh equiconvergence result and the universality properties of power series
Sorin G. Gal, Irene Sabadini

Chapter 6. Inequalities for Quaternionic Polynomials

Abstract
Many results in approximation theory depend on the fact that the derivative of a polynomial cannot be, in general, too large.
Sorin G. Gal, Irene Sabadini

Chapter 7. Approximation of nullsolutions of generalized Cauchy–Riemann operators

Abstract
In this chapter we present some results concerning approximation theory in the setting of nullsolutions of generalized Cauchy–Riemann operators in the quaternionic and in the Clifford algebra setting. This last case was considered only marginally in this work, but in this chapter we deal also with this more general situation when it is the framework of the original sources. It is obvious that one can always specialize the results to quaternions.
Sorin G. Gal, Irene Sabadini

Backmatter

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