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Lectures on entire functions
B. Ya. Levin

Translations of Mathematical Monographs
1996; 248 pp; softcover
Volume: 150
ISBN-10: 0-8218-0897-4
ISBN-13: 978-0-8218-0897-9
List Price: US$93
Member Price: US$74.40
Order Code: MMONO/150.S
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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.


Graduate students studying the theory of analytic functions and research mathematicians working in adjacent fields and applications.


"Readers with some previous knowledge will experience pleasant surprises at the deft way in which the material is presented. David Drasin's excellent translation is smooth and idiomatic. It reads like a book written in English by a gifted expositor."

-- Zentralblatt MATH

"These 28 lectures are written in the elegant form that was typical of the teaching style of B. Ya. Levin."

-- Mathematical Reviews

"Very welcome ... superbly organised ... would form an ideal basis for an advanced course. The student is not overloaded, but each lecture contains new and interesting material."

-- Bulletin of the London Mathematical Society

Table of Contents

Part I. Entire Functions of Finite Order
  • Growth of entire functions
  • Main integral formulas for functions analytic in a disk
  • Some applications of the Jensen formula
  • Factorization of entire functions of finite order
  • The connection between the growth of an entire function and the distribution of its zeros
  • Theorems of Phragmén and Lindelöf
  • Subharmonic functions
  • The indicator function
  • The Pólya Theorem
  • Applications of the Pólya Theorem
  • Lower bounds for analytic and subharmonic functions
  • Entire functions with zeros on a ray
  • Entire functions with zeros on a ray (continuation)
Part II. Entire Functions of Exponential Type
  • Integral representaiton of functions analytic in the half-plane
  • The Hayman Theorem
  • Functions of class \(C\) and their applications
  • Zeros of functions of class \(C\)
  • Completeness and minimality of system of exponential functions in \(L^2(0,a)\)
  • Interpolation by entire functions of exponential type
  • Interpolation by entire functions of the spaces \(L_\pi\) and \(B_\pi\)
  • Sin-type functions
  • Riesz bases formed by exponential functions in \(L^2(-\pi ,\pi )\)
  • Completeness of the eigenfunction system of a quadratic operator pencil
Part III. Some Additional Problems of the Theory of Entire Functions
  • Carleman's and R. Nevanlinna's formulas and their applications
  • Uniqueness problems for Fourier transforms and for infinitely-differentiable functions
  • The Matsaev Theorem on the growth of entire functions admitting a lower bound
  • Entire functions of the class \(P\)
  • S. N. Bernstein's inequality for entire functions of exponential type and its generalizations
  • Bibliography
  • Author index
  • Subject index
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