Entire Functions

Name: Entire Functions | Entire Functions
Brand: Academic Press
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Unknown Author(Herausgeber*in)

Academic Press

Erschienen am 29. August 2011

275 Seiten

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978-0-08-087313-8 (ISBN)

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Inhalt

Front Cover
Entire Functions
Copyright Page
Contents
Preface
Chapter 1. Introduction
1.1. Terminology
1.2. Jensen's, Carleman's, and Nevanlinna's formulas
1.3. Carathéodory's inequality
1.4. Phragmén-Lindelöf theorems
1.5. Density of sequences and sets
1.6. Stirling's formula.
1.7. Mittag-Leffler summability
1.8. Laplace and Fourier transforms
Chapter 2. General Properties of Entire Functions of Finite Order
2.1. Measures of rate of growth
2.2. Order and type in terms of the coefficients
2.3. Other connections between growth and coefficients
2.4. The order and type of a derivative
2.5. Rate of growth and distribution of zeros
2.6. Lemmas on infinite products
2.7. Hadamard's factorization theorem
2.8. Laguerre's theorem on separation of zeros
2.9. The zeros of functions of nonintegral order
2.10. The zeros of functions of integral order
2.11. Further relations between growth and zeros
2.12. Functions of genus 0
Chapter 3. The Minimum Modulus
3.1. Functions of order less than ½
3.2. Functions of order less than 1
3.3. Functions of order 1
3.4. The minimum modulus of a polynomial
3.5. Lemmas on functions of small order
3.6. Functions of order zero
3.7. Functions of larger order
Chapter 4. Functions with Real Negative Zeros
4.1. Direct theorems
4.2. Converse theorems
4.3. Generalizations
4.4. Another kind of theorem
Chapter 5. General Properties of Functions of Exponential
5.1. Properties of the indicator
5.2. Convex sets
5.3. The indicator diagram
5.4. Properties of the indicator diagram
5.5. The Borel transform on the boundary of the conjugate indicator diagram
5.6. Functions of exponential type in an angle
Chapter 6. Functions of Exponential Type. Restricted on a Line. I. Theormes in the Large
6.1. Introduction
6.2. Functions bounded on a line
6.3. Relations among integrals, mean values, and distribution of zeros
6.4. Generalizations of boundedness
6.5. Representation of functions in a half plane
6.6. Consequences of the representation theorem
6.7. Integrability on a line
6.8. Representations for functions which are bounded on a line
6.9. The indicator diagram of a finite Fourier transform
6.10. Periodic and almost periodic functions
Chapter 7. Functions of Exponential Type, Restricted on a Line. II. Asymptotic Behavior in a Half Plane
7.1. Introduction
7.2. The asymptotic behavior of a Blaschke product
7.3. Asymptotic behavior with other boundary conditions
7.4. The limit of r-1 log M ( r )
7.5. Factorization of a positive function
7.6. The type of a product
7.8. Functions with a zero-free half plane
Chapter 8. Functions of Exponential Type : Connections Between Growth and Distribution of Zeros
8.1. Introduction
8.2. Entire functions with real zeros
8.3. Entire functions with real zeros, continued
8.4. Entire functions with zeros close to the real axis
Chapter 9. Uniqueness Theorems
9.1. Introduction
9.2. Zeros at the positive integers
9.3. Applications of Carleman's theorem
9.4. Zeros at the positive and negative integers
9.5. General distribution of zeros on a half line
9.6. Zeros near the integers
9.7. Zeros less regularly distributed
9.8. Functions with a sequence of small values
9.9. Alternation theorems
9.10. Uniqueness theorems with operators
9.11. Generalized Abel series
9.12. Integral-valued entire functions
Chapter 10. Growth Theorems
10.1. Exponential growth on an arithmetic progression
10.2. Boundcdness on an arithmetic progression
10.3. Exponential growth on a sequence having a density
10.4. An application
10.5. Boundedness on a sequence which is close to an arithmetic progression
10.6. Applications to Lp and lp problems
10.7. Generalizations
Chapter 11. Operators and Their Extremal Properties
11.1. Introduction
11.2. A general method
11.3. Bounds for derivatives
11.4. Inequalities involving the maximum of I f(x) I
11.5. Inequalities involving sup I f (n p / r) I
11.6. Extension of operators
11.7. A class of operators
Chapter 12. Applications
12.1. Introduction
12.2. Asymptotic behavior of a mean value
12.3. A theorem on convolutions
12.4. Completeness of sets of functions
12.5. Fourier series
12.6. Power series on the circle of convergence
12.7. Dirichlet series
12.8. Gap theorems for entire functions
12.9. Expansions of analytic functions in series of polynomials
12.10. Differential equations of infinite order
12.11. Approximation by entire functions
Bibliography
Index
Index of Notations

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