Entire Functions (eBook)

Front Cover 1
Entire Functions 4
Copyright Page 5
Contents 10
Preface 8
Chapter 1. Introduction 14
1.1. Terminology 14
1.2. Jensen’s, Carleman’s, and Nevanlinna’s formulas 15
1.3. Carathéodory’s inequality 15
1.4. Phragmén-Lindelöf theorems 16
1.5. Density of sequences and sets 18
1.6. Stirling’s formula. 19
1.7. Mittag-Leffler summability 19
1.8. Laplace and Fourier transforms 20
Chapter 2. General Properties of Entire Functions of Finite Order 21
2.1. Measures of rate of growth 21
2.2. Order and type in terms of the coefficients 22
2.3. Other connections between growth and coefficients 25
2.4. The order and type of a derivative 26
2.5. Rate of growth and distribution of zeros 27
2.6. Lemmas on infinite products 31
2.7. Hadamard’s factorization theorem 35
2.8. Laguerre’s theorem on separation of zeros 36
2.9. The zeros of functions of nonintegral order 37
2.10. The zeros of functions of integral order 40
2.11. Further relations between growth and zeros 45
2.12. Functions of genus 0 47
Chapter 3. The Minimum Modulus 52
3.1. Functions of order less than ½ 52
3.2. Functions of order less than 1 53
3.3. Functions of order 1 56
3.4. The minimum modulus of a polynomial 59
3.5. Lemmas on functions of small order 60
3.6. Functions of order zero 63
3.7. Functions of larger order 64
Chapter 4. Functions with Real Negative Zeros 68
4.1. Direct theorems 68
4.2. Converse theorems 69
4.3. Generalizations 72
4.4. Another kind of theorem 73
Chapter 5. General Properties of Functions of Exponential 79
5.1. Properties of the indicator 79
5.2. Convex sets 83
5.3. The indicator diagram 86
5.4. Properties of the indicator diagram 88
5.5. The Borel transform on the boundary of the conjugate indicator diagram 90
5.6. Functions of exponential type in an angle 91
Chapter 6. Functions of Exponential Type. Restricted on a Line. I. Theormes in the Large 95
6.1. Introduction 95
6.2. Functions bounded on a line 95
6.3. Relations among integrals, mean values, and distribution of zeros 98
6.4. Generalizations of boundedness 102
6.5. Representation of functions in a half plane 104
6.6. Consequences of the representation theorem 108
6.7. Integrability on a line 110
6.8. Representations for functions which are bounded on a line 116
6.9. The indicator diagram of a finite Fourier transform 121
6.10. Periodic and almost periodic functions 122
Chapter 7. Functions of Exponential Type, Restricted on a Line. II. Asymptotic Behavior in a Half Plane 126
7.1. Introduction 126
7.2. The asymptotic behavior of a Blaschke product 127
7.3. Asymptotic behavior with other boundary conditions 134
7.4. The limit of r-1 log M ( r ) 136
7.5. Factorization of a positive function 137
7.6. The type of a product 139
7.8. Functions with a zero-free half plane 142
Chapter 8. Functions of Exponential Type : Connections Between Growth and Distribution of Zeros 146
8.1. Introduction 146
8.2. Entire functions with real zeros 149
8.3. Entire functions with real zeros, continued 151
8.4. Entire functions with zeros close to the real axis 156
Chapter 9. Uniqueness Theorems 165
9.1. Introduction 165
9.2. Zeros at the positive integers 166
9.3. Applications of Carleman’s theorem 167
9.4. Zeros at the positive and negative integers 169
9.5. General distribution of zeros on a half line 170
9.6. Zeros near the integers 174
9.7. Zeros less regularly distributed 177
9.8. Functions with a sequence of small values 178
9.9. Alternation theorems 181
9.10. Uniqueness theorems with operators 182
9.11. Generalized Abel series 185
9.12. Integral-valued entire functions 188
Chapter 10. Growth Theorems 191
10.1. Exponential growth on an arithmetic progression 191
10.2. Boundcdness on an arithmetic progression 193
10.3. Exponential growth on a sequence having a density 198
10.4. An application 203
10.5. Boundedness on a sequence which is close to an arithmetic progression 204
10.6. Applications to Lp and lp problems 209
10.7. Generalizations 212
Chapter 11. Operators and Their Extremal Properties 219
11.1. Introduction 219
11.2. A general method 220
11.3. Bounds for derivatives 223
11.4. Inequalities involving the maximum of I f(x) I 226
11.5. Inequalities involving sup I f (n p / r) I 231
11.6. Extension of operators 235
11.7. A class of operators 238
Chapter 12. Applications 246
12.1. Introduction 246
12.2. Asymptotic behavior of a mean value 246
12.3. A theorem on convolutions 247
12.4. Completeness of sets of functions 247
12.5. Fourier series 250
12.6. Power series on the circle of convergence 252
12.7. Dirichlet series 256
12.8. Gap theorems for entire functions 257
12.9. Expansions of analytic functions in series of polynomials 258
12.10. Differential equations of infinite order 260
12.11. Approximation by entire functions 261
Bibliography 265
Index 282
Index of Notations 289