This monograph deals with functions of completely regular growth (FCRG), i.e., functions that have, in some sense, good asymptotic behaviour out of an exceptional set. The theory of entire functions of completely regular growth of on variable, developed in the late 1930s, soon found applications in both mathematics and physics. Later, the theory was extended to functions in the half-plane, subharmonic functions in space, and entire functions of several variables. This volume describes this theory and presents recent developments based on the concept of weak convergence. This enables a unified approach and provides a comparatively simple presentation of the classical Levin-Pfluger theory. Emphasis is put on those classes of functions which are particularly important for applications functions having a bounded spectrum and finite exponential sums.
For research mathematicians and physicists whose work involves complex analysis and its applications. The book will also be useful to those working in some areas of radiophysics and optics.
Table of Contents1. Entire functions of completely regular growth of one variable.- §1. Preliminaries.- §2. Regularity of growth, D’-convergence and right distribution of zeros.- §3. Rays of completely regular growth. Addition of indicators.- Notes.- 2. Subharmonic functions of completely regular growth in Rn.- §1. General information on subharmonic functions. D*-convergence ..- §2. Criteria for regularity of growth in Rn.- §3. Rays of completely regular growth and limit sets.- §4. Addition of indicators.- Notes.- 3. Entire functions of completely regular growth in Cn.- §1. Functions of c completely regular growth on complex rays.- §2. Addition of indicators.- §3. Entire functions with prescribed behaviour at infinity.- Notes.- 4. Functions of completely regular growth in the half-plane or a cone.- §1. Preliminary information on functions holomorphic in a half-plane.- §2. Functions of completely regular growth in C+.- §3. Functions of completely regular growth in C+.- §4. Functions of completely regular growth in a cone.- Notes.- 5. Functions of exponential type and bounded on the real space (Fourier transforms of distribution of compact support).- §1. Regularity of growth of entire functions of exponential type and bounded on the real space.- §2. Discrete uniqueness sets.- §3. Norming sets.- Notes.- 6. Quasipolynomials.- §1. M-quasipolynomials. Growth and zero distribution.- §2. Entire functions that are quasipolynomials in every variable.- §3. Factors of quasipolynomials.- Notes.- 7. Mappings.- §1. Information on the general theory of holomorphic mappings.- §2. Plurisubharmonic functions of ?-regular growth and asymptotic behaviour of order functions of holomorphic mappings.- §3. Jessen’s theorem for almost periodic holomorphic mappings.- Notes.