Lectures on complex approximation

Gaier D. Lectures on complex approximation (Birkhauser, 1987)(ISBN 081763147X)(600dpi)(210s)\_MCat\_.zip(KA)-new Cover......Page 1 Title: Lectures on Complex Approximation......Page 2 ISBN 0-8176-3 I47-X......Page 3 Table of Contents......Page 4 Preface to the English Edition......Page 8 Preface to the German Edition......Page 10 SYMBOLS AND NOTATION......Page 12 PART I: APPROXIMTION BY SERIES EXPANSIONS AND BY INTERPOLATION......Page 14 §1. The Hilbert Space L^2(G)......Page 15 A1. The Gram-Schmidt orthogonolization process......Page 19 A2. The Gramian matrix......Page 20 A3 • A special case: Polynomials in L^2 (G)......Page 22 B. Zeros of orthogonal polynomials......Page 24 C. Asymptotic representation of the ON polynomials......Page 25 A. The problem and examples......Page 29 B. Domains with the PA property......Page 30 C2 • Moon-shaped domains......Page 33 A. ON expansions in Hilbert space......Page 37 B. ON expansions in the space L^2 (G)......Page 38 C. The uality of the approximation if f is analytic in G......Page 39 Remarks about §4......Page 42 A. Introduction and properties of the kernel function......Page 43 B. Series representation of the Bergman kemel function......Page 44 C. Constmction of confonnal mappings with the Bergman kernel function......Page 45 C1 • The connection between K and confonnal mapping......Page 46 c2 • The Bieberbach polynomials......Page 47 c3 . The use of singular functions in the ON process......Page 49 D2 • Representation of integral f(x)dx as an area integral......Page 50 §6. The quality of the approximation; Faber expansions......Page 53 A. Boundary behavior of Cauchy integrals......Page 54 B. Faber polynomials and Faber expansions......Page 55 c1 • Curves of bounded rotation......Page 58 c2. The Faber mapping T......Page 60 D1 • Preparations; unifonn convergence......Page 64 D2. The modulus of continuity of the Cauchy integral corresponding to h......Page 65 D3. The quality of the approximation......Page 66 E1. Additional uniform estimates......Page 68 E2 • Local estimates......Page 69 Remarks about §6......Page 70 A. The interpolating polynomial......Page 71 B. Special cases of Hennite's fonnula......Page 72 A. Preparations; rough statement about convergence......Page 75 B. General convergence theorem of Kalmar and Walsh......Page 77 C. The system of Feje'r points......Page 80 D. The system of Fekete points......Page 83 Remarks about §2......Page 85 A. Again: Interpolation in Fekete points......Page 86 B. Runge'8 approximation theorem......Page 89 A. Interpolation on {z: Izl = r}, r