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Intro; Preface; Contents; Chapter 1 Preliminaries on Hypercomplex Analysis; 1.1 An introduction to slice regular functions; 1.2 Slice regular functions, an alternative definition; 1.3 Further properties of slice regular functions; 1.4 Preliminary results on quaternionic polynomials; 1.5 Slice monogenic functions; 1.6 Other hypercomplex function theories; 1.7 A connection between theories of hyperholomorphy via the Fueter mapping theorem; Chapter 2 Approximation of Continuous Functions; 2.1 Weierstrass-Stone Type Results; 2.2 Carleman Type Results; 2.3 Notes on Müntz-Szász Type Results
Chapter 3 Approximation in Compact Balls by Bernstein and Convolution Type Operators3.1 Approximation by q-Bernstein Polynomials, q≥1, in Compact Balls; 3.2 Approximation in Compact Balls and in Cassini Cells; 3.2.1 Approximation by quaternionic polynomial convolutions; 3.2.2 Approximation by nonpolynomial quaternion convolutions; 3.2.3 Approximation by convolution operators of a paravector variable; Chapter 4 Approximation of Slice Regular Functions in Compact Sets; 4.1 Runge Type Results; 4.2 Mergelyan Type Results; 4.2.1 Riemann mappings for axially symmetric sets
4.2.2 Approximation in some particular compact sets4.3 Arakelian Type Results; 4.4 Approximation by Faber Type Polynomials; 4.4.1 Biholomorphic bijections between complements of compact axially symmetric sets; 4.4.2 Quaternionic Faber polynomials and Faber series; 4.5 Approximation by Polynomials in Bergman Spaces; 4.5.1 Approximation in Bergman spaces of first kind; 4.5.2 Approximation in Bergman spaces of second kind; Chapter 5 Overconvergence, Equiconvergence and Universality Properties; 5.1 Overconvergence of Chebyshev and Legendre Polynomials; 5.2 Walsh Equiconvergence Type Results
5.2.1 The interpolation problem5.2.2 Walsh equiconvergence theorem; 5.3 Universality Properties of Power Series and Entire Functions; 5.3.1 Almost universal power series; 5.3.2 Almost universal entire functions; Chapter 6 Inequalities for Quaternionic Polynomials; 6.1 The complex case; 6.2 Bernstein's Inequality; 6.3 Erdos-Lax's Inequality; 6.4 Turán's Inequality; 6.5 Notes on Turan Type Inequalities; Chapter 7 Approximation of nullsolutions of generalized Cauchy-Riemann operators; 7.1 Runge Type Results; 7.2 Appell systems; 7.3 Approximation in L2
7.4 Approximation of axially monogenic functions7.5 Series expansions of monogenic functions; 7.6 Polynomial approximation of monogenic quasiconformal maps; Bibliography; Index
Chapter 3 Approximation in Compact Balls by Bernstein and Convolution Type Operators3.1 Approximation by q-Bernstein Polynomials, q≥1, in Compact Balls; 3.2 Approximation in Compact Balls and in Cassini Cells; 3.2.1 Approximation by quaternionic polynomial convolutions; 3.2.2 Approximation by nonpolynomial quaternion convolutions; 3.2.3 Approximation by convolution operators of a paravector variable; Chapter 4 Approximation of Slice Regular Functions in Compact Sets; 4.1 Runge Type Results; 4.2 Mergelyan Type Results; 4.2.1 Riemann mappings for axially symmetric sets
4.2.2 Approximation in some particular compact sets4.3 Arakelian Type Results; 4.4 Approximation by Faber Type Polynomials; 4.4.1 Biholomorphic bijections between complements of compact axially symmetric sets; 4.4.2 Quaternionic Faber polynomials and Faber series; 4.5 Approximation by Polynomials in Bergman Spaces; 4.5.1 Approximation in Bergman spaces of first kind; 4.5.2 Approximation in Bergman spaces of second kind; Chapter 5 Overconvergence, Equiconvergence and Universality Properties; 5.1 Overconvergence of Chebyshev and Legendre Polynomials; 5.2 Walsh Equiconvergence Type Results
5.2.1 The interpolation problem5.2.2 Walsh equiconvergence theorem; 5.3 Universality Properties of Power Series and Entire Functions; 5.3.1 Almost universal power series; 5.3.2 Almost universal entire functions; Chapter 6 Inequalities for Quaternionic Polynomials; 6.1 The complex case; 6.2 Bernstein's Inequality; 6.3 Erdos-Lax's Inequality; 6.4 Turán's Inequality; 6.5 Notes on Turan Type Inequalities; Chapter 7 Approximation of nullsolutions of generalized Cauchy-Riemann operators; 7.1 Runge Type Results; 7.2 Appell systems; 7.3 Approximation in L2
7.4 Approximation of axially monogenic functions7.5 Series expansions of monogenic functions; 7.6 Polynomial approximation of monogenic quasiconformal maps; Bibliography; Index