Abstract
This is a survey of M. Krein's results in the theory of entire and meromorphic functions. Their further development is also discussed.
Similar content being viewed by others
References
M. G. Krein, “On a special class of entire and meromorphic functions,” in: N. I. Akhiezer and M. G. Krein,On Some Problems in the Theory of Moments [in Russian], GONTI, Khar'kov (1938), pp. 208–252.
M. G. Krein, “On representations of functions by Fourier-Stieltjes integrals,”Uchen. Zapiski Kuibyshev. Pedagog. Inst., Issue 7, 123–147 (1943).
M. G. Krein and A. A. Nudel'man, “On representations of entire functions positive on the real axis, on a semiaxis, or outside a finite interval,”Mat. Issled., Issue 61, 40–59 (1981).
M. G. Krein, B. Ya. Levin, and A. A. Nudel'man, “On special representations of polynomials that are positive on a system of closed intervals and some applications,” in: L. Liefman (editor),Functional Analysis, Optimization, and Mathematical Economics, Oxford University Press, Oxford-New York (1990), pp. 56–114.
M. G. Krein, “To the theory of entire functions of exponential type,”Izv. Akad. Nauk SSSR, Ser. Mat.,11, No. 4, 309–326 (1947).
M. G. Krein, “To the theory of entire matrix functions of exponential type,”Ukr. Mat. Zh.,3, No. 2, 164–173 (1951).
M. G. Krein, “On the indefinite case of the Sturm-Liouville boundary-value problem on the interval (0, τ),”Izv. Akad. Nauk SSSR, Ser. Mat.,16, Issue 4, 293–324 (1952).
M. G. Krein and B. Ya. Levin, “On entire almost periodic functions of exponential type,”Dokl. Akad. Nauk SSSR,64, Issue 3, 258–287 (1949).
B. Ya. Levin,Distributions of Roots of Entire Functions [in Russian], Gostekhteoretizdat, Moscow (1956).
N. G. Chebotarev and N. N. Meiman, “The Routh-Hurwitz problem for polynomials and entire functions,”Trudy Mat. Inst. Akad. Nauk SSSR,26, 1–332 (1949).
N. I. Akhiezer and B. Ya. Levin, “A generalization of the inequality of S. N. Bernshtein for derivatives of entire functions,” in:Studies in Modem Problems of the Theory of Functions of Complex Variable [in Russian], Fizmatgiz, Moscow (1960), pp. 125–183.
B. Ya. Levin, “Majorants in the classes of subharmonic functions,”Tear. Funkts., Funkts. Anal. Prilozhen., Issue 51, 3–17: Issue 52, 3–33(1989).
B. Ya. Levin, “Completeness of the systems of functions, quasianalyticity, and subharmonic majorants,”Zap. Nauchn. Semin. Leningrad. Old. Mat. Inst. Akad. Nauk SSSR,170, 102–156 (1989).
B. Ya. Levin and I. V. Ostrovskii, “On a dependence of the growth of an entire function on the location of zeros of its derivatives,”Sib. Mat. Zh.,1, No. 3, 427–455 (1960).
S. Hellerstein and J. Williamson, “Derivatives of entire functions and a question of Polya,”Trans. Am. Math. Soc.,227, 227–249 (1977);234, 497–503.
T. Sheil-Small, “On the zeros of the derivatives of real entire functions and Wiman's conjecture,”Ann. Math.,129, 179–193 (1989).
V. P. Gurarii, “Fourier transformation inL 2ϕ (−8, ∞) with a weight,”Mat. Sb.,58, No. 4, 439–452 (1962).
B. Ya. Levin.Entire Functions [in Russian], Moscow University, Moscow (1971).
A. A. GoI'dberg and I. V. Ostrovskii,Distributions of Values of Meromorphic Functions [in Russian], Nauka, Moscow (1970).
V. I. Matsaev, “On the growth of entire functions admitting certain lower bounds,”Dokl. Akad. Nauk SSSR,132, No. 2, 283–286 (1960).
E. N. Sergienko, “On the growth of meromorphic functions admitting a special lower bound,”Tear. Funk., Funkts. Anal. Prilozhen., Issue 21, 83–104 (1974).
E. N. Sergienko, “On the growth of functions representable as a difference of subharmonic functions and admitting a special lower bound,”Teor. Funk., Funkts. Anal. Prilozhen., Issue 37, 116–122 (1982).
W. K. Hayman and P. B. Kennedy,Subharmonic Functions [Russian translation]. Mir, Moscow (1980).
V. A. Marchenko and I. V. Ostrovskii, “Characteristics of the spectrum of the Hill operator,”Mat. Sb.,97, No. 4, 540–606 (1975).
I. V. Ostrovskii, “On a class of entire functions,”Dokl. Akad. Nauk SSSR,229, No. 1, 39–42 (1976).
V. E. Katsnel'son, “To the theory of entire functions from the Cartwright class,”Teor. Funk., Funkts. Anal. Prilozhen., Issue 42, 57–62 (1984).
B. G. Freidin, “On entire functions admitting a special lower bound,”Teor. Funk., Funkts. Anal. Prilozhen., Issue 46, 139–141 (1986).
A. A. GoI'dberg, “To a theorem of M. G. Krein,”Dokl. Soobshch. Uzhgorod. Univ., Ser. Fiz.-Mat. Nauk, No. 4, 106–108 (1961).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 87–99, January–February, 1994.
Rights and permissions
About this article
Cite this article
Ostrovskii, I.V. M. G. Krein's investigations in the theory of entire and meromorphic functions and their further development. Ukr Math J 46, 87–100 (1994). https://doi.org/10.1007/BF01057003
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01057003