M. G. Krein's investigations in the theory of entire and meromorphic functions and their further development
type: Item
instance of: scholarly article
title: M. G. Krein's investigations in the theory of entire and meromorphic functions and their further development
zbMATH Open document ID: 0870.30001
DOI: 10.1007/BF01057003
published in: Ukrainian Mathematical Journal
publication date: 1995-11-02T00:00:00Z
review text: This is a comprehensive survey of the results of Mark G. Krein (1907-1989) in the theory of entire and meromorphic functions. The results were published in eight articles (some of them are joint works with B. Ya. Levin and A. A. Nudel'man) in 1938-1990. I. V. Ostrovskij classified the Krein's results into four groups as follows: 1. The generalization to entire functions of the Hurwitz criterion for a polynomial to have all the roots in the left half-plane, and also the generalization of the Hermite-Biehler theorem. 2. The generalization to entire functions of the Fejér-Riez theorem on the representation of a trigonometric polynomial, which is nonnegative on the real axis, as a square of the modulus of an another polynomial. 3. Investigation of entire functions from the so-called Cartwright class, i.e., those satisfying the condition \(\int^\infty_{-\infty}{\ln^+|f(t)|\over 1+t^2} dt<\infty\). 4. Distribution of the zeros of entire functions, which are almost-periodic in the H. Bohr's sense and have bounded spectra. The author gave detailed description of Krein's results related to the first three topics, and state-of-the-art presentation of the further development.
reviewed by: Alexander I. Kheyfits
Mathematics Subject Classification ID: 30-02
Mathematics Subject Classification ID: 30D99
zbMATH DE Number: 803820
zbMATH Keywords: entire functions
author: I. V. Ostrovskii
MaRDI profile type: Publication
cites work: Q5790967
cites work: Q5801454
cites work: Q3478655
cites work: Derivatives of Entire Functions and A Question of Polya
cites work: On the zeros of the derivatives of real entire functions and Wiman's conjecture
cites work: Q3671464
cites work: Q4092300
cites work: Q4132799
description: scientific article; zbMATH DE number 803820