Distribution of Zeros of Entire Functions (Translations of Mathematical Monographs)

Cover page......Page 1
Title page......Page 2
Contents......Page 4
PREFACE TO THE SECOND EDITION......Page 8
PREFACE TO THE FIRST EDITION......Page 10
1.  The growth scale......Page 12
2.  The connection between the growth of an entire function and the rate of decrease of its Taylor coefficients......Page 15
3.  The expansion of entire functions in infinite products......Page 17
4.  Estimates for canonical products......Page 20
5.  Jensen\'s theorem......Page 25
6.  The relation between the maximum modulus of a holomorphic function and the maximum of its real part......Page 28
7.  Lower bounds for the modulus of a polynomial......Page 30
8.  Lower bounds for the modulus of a holomorphic function......Page 32
9.  The growth of the product of two entire functions......Page 33
10.  Hadamard\'s theorem......Page 35
11.  Entire functions of integral order......Page 38
12.  Proximate orders......Page 42
13.  Extension of the classical theorems to proximate orders......Page 52
14.  The Phragmen-Lindelof principle......Page 58
15.  The indicator function......Page 62
16.  The fundamental relations and analytic properties of the indicator function......Page 64
17.  Auxiliary functions......Page 72
18.  The generalized indicator......Page 81
19.  Plane convex sets......Page 86
20.  Entire functions of exponential type......Page 96
1.  Basic results......Page 101
2.  Entire functions of nonintegral order with regular distribution of zeros (proof of theorem 1)......Page 109
3.  Entire functions of integral order with regular distribution of zeros (proof of theorem 2)......Page 120
4.  Construction of an entire function with a given indicator (proof of theorem 3)......Page 128
5.  Asymptotic representation of entire functions with regular distribution of zeros (proof of theorem 4)......Page 134
6.  Entire functions with regular sets of zeros (proof of theorem 5)......Page 137
7.  Theorems on equicontinuity (proof of theorems 6 and 7)......Page 140
CHAPTER 3 - FUNCTIONS OF COMPLETELY REGULAR GROWTH......Page 151
1.  Set of rays of completely regular growth......Page 152
2.  A generalized formula of Jensen. Investigation of the function J}(#)......Page 154
3.  The basic theorem on functions of completely regular order......Page 164
4.  Indicator of the product of two functions......Page 171
5.  Some consequences of the generalized formula of Jensen. Case of a sinusoidal indicator......Page 173
CHAPTER 4 - UNIQUENESS, INTERPOLATION AND COMPLETENESS......Page 180
1.  Uniqueness theorems for entire functions of finite order......Page 182
2.  Uniqueness theorems for functions of finite order, holomorphic in an angle......Page 186
3.  Functions vanishing on a set with an angular density......Page 202
4.  Representation of entire functions of Lagrange interpolation series......Page 206
5.  Some applications of Lagrange interpolation series......Page 214
6.  Completeness of systems of functions. The connection between completeness and uniqueness......Page 222
7.  Theorems on the completeness of some systems of entire functions......Page 226
CHAPTER 5 - FUNCTIONS OF CLASS A......Page 234
1.  Formula of Carleman. Criterion for an entire function of exponential type to be of class A......Page 235
2.  Representation of a function harmonic in a half-plane......Page 242
3.  Representation of a function of exponential type and of class A in the upper half-plane......Page 248
4.  Functions of class A and of completely regular growth......Page 255
5.  Indicator diagram of an entire function of exponential type, and of class A......Page 264
6.  Theorem of M. G. Krein on the expansion of the reciprocal of an entire function......Page 270
1.  Information from the theory of almost-periodic functions......Page 276
2.  Roots of an almost-periodic function with a bounded spectrum......Page 280
3.  A general theorem on roots and on mean motion for holomorphic almost-periodic functions......Page 285
4.  A theorem on mean motion for an almost-periodic function with semibounded spectrum......Page 291
5.  Functions approximated by exponential polynomials......Page 299
6.  Growth of a function of class E_I for a norming region in the form of a polygon......Page 306
7.  Growth of a function of class E_I outside an arbitrary norming region I......Page 313
CHAPTER 7 - THE THEOREM OF HERMITE-BIEHLER FOR ENTIRE FUNCTIONS......Page 317
1.  Representation of a real meromorphic function mapping the upper half-plane onto the upper half-plane......Page 319
2.  Generalization of the theorem of Hermite-Biehler to arbitrary entire functions......Page 323
3.  Representation of a function of class HB......Page 329
4.  The Hermite-Biehler theorem for entire functions of exponential type......Page 330
1.  Functions which can be approximated by polynomials, all of whose zeros lie in an angle......Page 339
2.  Theorems on composition of polynomials......Page 348
3.  Multiplier sequences......Page 352
CHAPTER 9 - OPERATORS PRESERVING INEQUALITIES AMONG ENTIRE FUNCTIONS......Page 360
1.  Majorants and admissible classes......Page 362
2.  Some properties of the class P*......Page 364
3.  Operators preserving subordination (\\mathcal{B}_I-operators)......Page 367
4.  The class P and inequalities on the real axis......Page 374
5.  Classes of functions of several variables......Page 378
6.  General form of the operators \\mathcal{B} and \\mathcal{B}*......Page 385
7.  Certain extremal properties of entire functions......Page 392
1.  The impossibility of constructing an exact. scale of growth......Page 395
2.  Convergent and divergent types......Page 396
3.  The Paley-Wiener Theorem......Page 398
4.  Levitan polynomials......Page 404
5.  Power series with coefficients that are the values of entire functions of exponential type for integral values of their arguments......Page 405
1.  Functions defined by their values on an interval......Page 408
2.  Quasianalytic classes of almost-periodic functions......Page 418
1.  Completeness theorems......Page 425
2.  Minireality (strong linear independence) of the system \\{ e^{i\\lambda_k x} \\}......Page 432
1.  Completeness of a system of solutions of a differential equation of second order in regions in the complex plane......Page 437
2.  The inverse problem for the Sturm-Liouville equation......Page 443
APPENDIX 5 - REPRESENTATION OF A POSITIVE ENTIRE FUNCTION OF EXPONENTIAL TYPE AS THE SQUARE OF THE ABSOLUTE VALUE OF AN ENTIRE FUNCTION......Page 450
1.  An expression for the real part of a zero of a function of the class [\\Delta]......Page 457
2.  A characterization of the set of zeros of an almost-periodic function of the class [\\Delta]......Page 462
3.  The relation between the Fourier series \\psi(x) and f(x)......Page 472
APPENDIX 7 - MISCELLANEOUS THEOREMS AND PROBLEMS......Page 477
1.  Subharmonic functions......Page 483
2.  Functions of completely regular growth in an angle......Page 485
3.  Regularity of the mass distribution and exceptional sets......Page 488
4.  The second term of an asymptotic formula......Page 491
5.  Derivatives and primitives of entire functions of completely regular growth......Page 493
6.  Some complements to the theory of entire functions of integral order......Page 495
7.  Upper and lower indicators and their applications to the distribution of the zeros with respect to their arguments......Page 497
8.  Miscellaneous results from the theory of functions of completely regular growth......Page 502
LIST OF IMPORTANT IDEAS AND THEOREMS......Page 505
BIBLIOGRAPHY......Page 521