M. G. Krein's investigations in the theory of entire and meromorphic functions and their further development
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.
scientific article; zbMATH DE number 803820