Lectures on Complex Approximation

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 <1......Page 91
B. Interpolation on {z: Izl = I}......Page 94
C. Approximation by rational functions......Page 99
Rem8lks about 4......Page 100
PART II: GENERAL APPROXIMATION THEOREMS IN THE COMPLEX PLAINE......Page 104
§1. Runge\'s approximation theorem......Page 105
A. General Cauchy fonnula......Page 106
B. Runge\'s theorem......Page 107
C. The pole shifting method......Page 108
A. Fonnulation of the result; special cases; consequences......Page 110
B. Preparations for the proof......Page 111
B2. A representation formula......Page 112
B3 •  Koehe\'s 1/4-theorem......Page 113
B4• Mergelyan\'s lemma......Page 114
C. Proof of mergelyan\'s theorem......Page 117
§3. Apptoximation by rational functions......Page 122
A1. Alice Roth\'s construction......Page 123
A2• Swiss cheese with interior points......Page 125
A4. Accumulation of holes along the diameter of D......Page 126
B1. An integral transfonn......Page 127
B2 • Partition of unity......Page 128
C1. The localization theorem......Page 129
C2.  Applications of Bishop\'s theorem......Page 131
D. Vitushkin\'s theorem; a report......Page 134
§4. Roth\'s fusion lemma......Page 135
A. The fusion lemma......Page 136
B. A new proof of Bishop\'s theorem......Page 140
Remark about §4......Page 142
A. Statement of the problem......Page 143
B. Roth\'s approximation theorem......Page 144
C1• The one-point compactification G* of G; connectedness of G* \\F......Page 146
C2. Three sufficient criteria for meromorphic approximation......Page 147
§2. Uniform approximation by analytic functions......Page 149
A. Moving the poles of meromorphic functions......Page 150
B. Preliminary topological remarks......Page 151
C. Arakeljan\'s approximation theorem......Page 152
C1. Approximation of meromorphic functions by analytic functions......Page 153
C2. Arakeljan\'s theorem......Page 155
Remarks about §2......Page 157
A1. Tangential approximation; e-approximation......Page 158
A2. Two lemmas......Page 159
A3. Carleman\'s theorem......Page 162
B1 • Sufficient conditions for e-approximation......Page 164
C1. Condition (A); a lemma......Page 168
C2 • Nersesjan theorem......Page 170
Remarks about §3......Page 172
§4. Approximation with certain error functions......Page 173
A. e-approxunatlon without condition (A)......Page 174
B. Growth of the approximating function......Page 175
C. The special case F = R......Page 176
A. Radial boundary values of entire functions......Page 177
B1. A general approximation theorem......Page 181
B2 • The Dirichlet problem for radial limits......Page 183
C. Approximation and uniqueness theorems......Page 184
D1 • Prescribed boundary behavior along countably many curves......Page 186
D2 • Analytic functions with prescribed cluster sets......Page 187
D4. Julia directions of entire functions......Page 188
Remarks about §5......Page 189
References......Page 191
INDEX......Page 206
Back Cover......Page 210