Advances in hypercomplex analysis

✍ Scribed by Graziano Gentili; et al (eds.)

• Publisher: Springer
• Year: 2013
• Tongue: English
• Leaves: 149
• Series: Springer INdAM series, v. 1
• Category: Library

✦ Synopsis

The work aims at bringing together international leading specialists in the field of Quaternionic and Clifford Analysis, as well as young researchers interested in the subject, with the idea of presenting and discussing recent results, analyzing new trends and techniques in the area and, in general, of promoting scientific collaboration. Particular attention is paid to the presentation of different notions of regularity for functions of hypercomplex variables, and to the study of the main features of the theories that they originate. Read more... Regular vs. Classical Möbius Transformations of the Quaternionic Unit Ball / Cinzia Bisi and Caterina Stoppato -- Distributional Boundary Values of Harmonic Potentials in Euclidean Half-Space as Fundamental Solutions of Convolution Operators in Clifford Analysis / Fred Brackx, Hendrik De Bie and Hennie De Schepper -- On Two Approaches to the Bergman Theory for Slice Regular Functions / Fabrizio Colombo, J. Oscar González-Cervantes, Maria Elena Luna-Elizarrarás, Irene Sabadini and Michael Shapiro -- A Bloch-Landau Theorem for Slice Regular Functions / Chiara Della Rocchetta, Graziano Gentili and Giulia Sarfatti -- Dirichlet-Type Problems for the Iterated Dirac Operator on the Unit Ball in Clifford Analysis / Min Ku, Uwe Kähler and Paula Cerejeiras -- Fueter Regularity and Slice Regularity: Meeting Points for Two Function Theories / Alessandro Perotti -- A Note on Analytic Functionals on the Complex Light Cone / Daniele C. Struppa -- The S-spectrum for Some Classes of Matrices / Mihaela B. Vajiac -- Regular Composition for Slice-Regular Functions of Quaternionic Variable

✦ Table of Contents

Cover......Page 1
Advances in Hypercomplex Analysis......Page 3
Preface......Page 5
Contents......Page 6
1 Classical Möbius Transformations and Poincaré Distance on the Quaternionic Unit Ball......Page 7
2 Regular Fractional Transformations......Page 10
3 Regular Möbius Transformations of B......Page 13
4 Differential and Metric Properties of Regular Möbius Transformations......Page 14
5 The Schwarz-Pick Lemma for Regular Functions......Page 17
References......Page 19
1 Introduction......Page 20
2 Basics of Clifford Analysis......Page 22
3 Harmonic and Monogenic Potentials in R+m+1......Page 24
4 Powers of the Dirac Operator......Page 30
5 A New Operator......Page 33
6 Powers of the Laplace Operator......Page 37
7 A Second New Operator......Page 38
8 Conclusion......Page 41
References......Page 42
On Two Approaches to the Bergman Theory for Slice Regular Functions......Page 43
1 Introduction......Page 44
2 Preliminary Results on Slice Regular Functions......Page 45
3 The Slice Regular Bergman Theory of the First Kind......Page 46
4 The Slice Regular Bergman Kernel of the First Kind......Page 48
5 The Slice Regular Bergman Theory of the Second Kind......Page 53
References......Page 57
1 Introduction......Page 59
2 Preliminaries......Page 60
3 Uniform Norm and Regular Conjugation......Page 65
4 A Norm for a Mean Value Theorem......Page 67
5 The Bloch-Landau Type Theorem......Page 71
References......Page 78
1 Introduction......Page 79
2 Preliminaries and Notations......Page 80
3 Several Lemmas......Page 81
4 Half Dirichlet Problems for Null-Solutions to Dk......Page 83
5 Dirichlet Problems for Poly-harmonic Functions......Page 90
References......Page 95
1 Introduction......Page 97
2 Fueter Regularity and Slice Regularity......Page 99
2.1 Fueter Regular Functions......Page 100
2.1.1 Holomorphic Functions w.r.t. a Complex Structure Jp......Page 101
2.3 Quaternionic Slice Regular Functions......Page 103
2.4.1 Computation of D(qm)......Page 105
2.4.2 Regular Extension of Slice-Regular Functions......Page 107
2.4.4 Characterization of Reg(SR(OmegaD)) in R(OmegaD)......Page 108
3.1 The Operators Dn......Page 111
3.2 Slice Regularity and the Full Dirac Operators......Page 114
3.3 The Case of D3......Page 118
3.4 The Space F(Omega)......Page 119
References......Page 120
1 Introduction and Notations......Page 122
2 Ehrenpreis' Fundamental Principle and the Duality Theorems......Page 123
3 Complex Laplacian and Bicomplex Numbers......Page 125
References......Page 126
1 Introduction......Page 128
2 Notions of Functional Calculus......Page 129
3.1 Two 2x2 Triangular Matrices......Page 132
3.3 kxk Triangular Matrices......Page 133
4.1 Special Symmetric Matrices......Page 134
4.2 General Symmetric Matrices......Page 135
5.1 The Jordan Matrix......Page 136
5.2 A Special Resolvent Matrix......Page 138
5.3 The Hankel Matrix......Page 139
5.4 Pauli Matrices......Page 140
References......Page 141
1 Introduction to Slice-Regularity in H......Page 143
2 Non-commutative Bell Polynomials and Slice-Regular Composition......Page 145
References......Page 148