As a brilliant university lecturer, B. Ya. Levin attracted a large audience of working mathematicians and of students from various levels and backgrounds. For approximately 40 years, his Kharkov University seminar was a school for mathematicians working in analysis and a center for active research.
This monograph aims to expose the main facts of the theory of entire functions and to give their applications in real and functional analysis. The general theory starts with the fundamental results on the growth of entire functions of finite order, their factorization according to the Hadamard theorem, properties of indicator and theorems of Phragmén-Lindelöf type.
Author(s): B. Ya. Levin, Yu Lyubarskii, M. Sodin, V. Tkachenko
Series: Translations of Mathematical Monographs
Publisher: American Mathematical Society
Year: 1996
Language: English
Pages: 242
Frontmatter......Page 1
Boris Yakovlevich Levin, 1906-1993......Page 2
Title......Page 3
Copyright......Page 4
Table of Contents......Page 5
Preface......Page 9
Introduction......Page 13
Part I. Entire Functions of Finite Order......Page 14
1.2 Order and type of entire functions......Page 16
1.3 The relation between the growth of an entire function...........Page 18
2.2 The Poisson-Jensen formula......Page 21
2.3 The Jensen formula......Page 22
2.4 The Nevanlinna characteristics......Page 23
2.5 Some corollaries of the Jensen formula......Page 25
3.1 A theorem on (I)-quasianalyticity......Page 26
3.2 The convergence exponent and the upper density..........Page 28
3.3 Completeness of a system of exponential functions......Page 30
3.4 Completeness of a special system of functions in countably..........Page 31
4.1 The Weierstrass canonical product......Page 35
4.2 The Hadamard theorem......Page 36
4.3 Estimates for canonical products......Page 38
5.1 Functions of noninteger order......Page 41
5.2 Functions of integer order......Page 42
6.1 Functions analytic inside an angle......Page 46
6.2 Entire functions with values in Banach algebras......Page 49
6.3 Applications of the Phragmen and Lindelof theorems..........Page 52
7.1 Definition and basic properties......Page 54
7.2 The F. Riesz theorem and the Jensen formula......Page 57
7.3 Phragmen-Lindelof theorems for subharmonic functions......Page 58
7.4 Logarithmically subharmonic functions......Page 59
8.1 The definition and p[rho]-trigonometric convexity of the indicator......Page 61
8.2 Properties of trigonometrically convex functions......Page 63
8.3 Applications of properties of the indicator function......Page 66
9.1 Supporting functions of convex sets......Page 70
9.2 The Borel transform and the Polya theorem......Page 72
10.1 The Paley-Wiener theorem......Page 76
10.2 Analytic continuation of a power series......Page 77
10.3 Analytic functionals......Page 80
11.1 The Caratheodory inequality......Page 82
11.2 The Cartan estimate......Page 83
11.3 Lower bounds for the modulus of an analytic function in a disk......Page 86
12.1 Asymptotic behavior of canonical products......Page 88
12.2 Theorem on a segment on the boundary of the indicator diagram......Page 90
12.3 Lower bound for the canonical product with positive zeros.........Page 93
13.1 The Valiron theorem......Page 98
13.2 Functions of completely regular growth......Page 101
Part II. Entire Functions of Exponential Type......Page 104
14.1 The R. Nevanlinna formula......Page 105
14.2 Representation of a function f(z) analytic in the half-plane..........Page 107
14.3 Application to the theory of quasianalytic classes......Page 111
Lecture 15. The Hayman Theorem......Page 114
16.1 Properties of functions of class C......Page 119
16.2 The Titchmarsh convolution theorem and a problem of Gelfand......Page 123
16.3 Mean periodic functions......Page 125
17.1 The generalized Jensen formula......Page 129
17.2 Asymptotic properties of zeros of functions of class C......Page 130
Lecture 18. Completeness and Minimality of Systems of Exponential Functions...........Page 135
19.1 Definition and basic properties......Page 140
19.2 Boundary values of functions of H^{p}_{+}......Page 142
19.3 M. Riesz's theorem on conjugate harmonic functions...........Page 145
19.4 The Paley-Wiener theorem for H^{2}_{+}......Page 149
20.1 Spaces L^{p}_{sigma} and B_{sigma}......Page 151
20.2 Interpolation theorem with integer nodes......Page 152
20.3 Interpolation in the spaces L^{p}_{pi}, 1 < p < {infinity}..........Page 153
21.1 Interpolation by functions from B_{pi} and L_{pi}......Page 156
21.2 Interpolation by functions from L^{p}_{sigma} with {sigma} < {pi}......Page 161
21.3 Interpolation in a class of entire functions......Page 163
22.1 Interpolation with nodes at the zeros of sine-type function......Page 164
22.2 Functions whose zeros are close to the integers......Page 167
23.1 Definition and properties of Riesz bases......Page 170
23.2 The 1/4-theorem......Page 173
A1. Twofold completeness of the system K_{a}......Page 181
A2. Completeness of the system K^{+}_{a}......Page 183
Part III. Some Additional Problems of the Theory of Entire Functions......Page 185
24.1 The Carleman formula......Page 186
24.2 The Phragmen-Lindelof principle as formulated ..........Page 189
24.3 R. Nevanlinna's formula for a half-disk......Page 191
25.1 Uniqueness theorem for Fourier transforms......Page 193
25.2 Construction of entire functions decaying on the real axis......Page 197
25.3 Uniqueness problem of Gelfand and Shilov for infinitely differentiable..........Page 202
26.1 A lower bound for harmonic functions of order greater than one..........Page 206
26.2 Refinement of the upper bound......Page 209
26.3 Proof of Matsaev's theorem......Page 210
26.4 Entire functions admitting a lower bound for {rho} <= 1......Page 211
27.1 Properties of functions of class P......Page 213
27.2 Meromorphic functions with interlacing zeros and poles......Page 216
27.3 Theorem of Hermite and Biehler for entire functions of exponential type......Page 218
28.1 P-majorants......Page 222
28.2 Operators preserving inequalities......Page 225
28.3 S.N. Bernstein's inequality and Banach algebras......Page 231
Added in Proof......Page 233
Bibliography......Page 234
Author Index......Page 239
Subject Index......Page 241
