Ukrainian Mathematical Journal

, Volume 46, Issue 1–2, pp 62–74 | Cite as

On M. G. Krein's contribution to the moment problem

  • A. A. Nudel'man


This is a survey of M. G. Krein' s papers devoted to the moment problem.


Moment Problem 
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  1. 1.
    Henry Landau (editor),Moments in Mathematics, Proceedings of Symposia in Applied Mathematics, Vol. 37, American Mathematical Society, Providence, RI (1987).Google Scholar
  2. 2.
    N. I. Akhiezer and M. G. Krein,On Some Problems in the Theory of Moments [in Russian], GONTI, Kharkov (1938).Google Scholar
  3. 3.
    A. A. Markov,New Applications of Continued Fractions. Selected Works [in Russian], Gostekhizdat, Moscow (1948).Google Scholar
  4. 4.
    P. L. Chebyshev, “On the interpolation in the case of a large number of observations,” in:Complete Works [in Russian], Izd. Akad. Nauk SSSR, Moscow (1948).Google Scholar
  5. 5.
    M. G. Krein and A. A. Nudel'man. “On representations of entire functions positive on the real axis, or on a semiaxis, or outside a finite interval,”Mat. Issled., Issue 61. 40–59 (1981).Google Scholar
  6. 6.
    M. G. Krein, “On a generalization of the investigations of Academician Markov on limiting values of integrals.” in:Proceedings of the Second All-Union Mathematical Congress [in Russian], Vol. 2, (1934), pp. 152–154.Google Scholar
  7. 7.
    M. G. Krein, “On positive additive functional in linear normed spaces,”Zapiski Mat. Obshch.,4, No. 14, 227–237 (1937).Google Scholar
  8. 8.
    M. G. Krein, “The ideas of P. L. Chebyshev and A. A. Markov in the theory of limiting values of integrals and their further development,”Usp. Mat. Nauk,6, Issue 4, 3–120 (1951).Google Scholar
  9. 9.
    S. Karlin and W. Studden,Tchebycheff Systems: with Application in Analysis and Statistics, Interscience (1966).Google Scholar
  10. 10.
    M. G. Krein and A. A. Nudel'man,Markov Moment Problem and Extremal Problems [in Russian], Nauka, Moscow (1973).Google Scholar
  11. 11.
    M. G. Krein and A. A. Nudel'man, “On an interpolational problem related to the Stieltjes moment problem,”Dokl. Akad. Nauk Ukr. SSSR, No. 12, 1068–1072 (1977).Google Scholar
  12. 12.
    A. A. Nudel'man and A. V. Tulub, “On extremal problems that appear in the theory of Van der Waals forces,”Tear. Mat. Fiz.,39, No. 3, 359–366 (1979).Google Scholar
  13. 13.
    I. G. Kaplan,Introduction to the Theory of Intermolecular Interactions [in Russian], Nauka, Moscow (1982).Google Scholar
  14. 14.
    P. L. Chebyshev, “Expansion of series ordered in the descending order of powers of a variable in continued fractions,” in:Complete Works [in Russian], Izd. Akad. Nauk SSSR, Moscow (1948).Google Scholar
  15. 15.
    F. P. Vasil'ev, A. Z. Ishmukhamedov, and M. M. Potapov,A Generalized Method of Moments in the Problems of Optimal Control [in Russian], Moscow University, Moscow, (1989).Google Scholar
  16. 16.
    N. I. Akhiezer and M. G. Krein. “An extremum problem for polynomials,”Ann. Univ. Sci. Budapest, Sec. Math.,3/4, 9–14 (1960/61).Google Scholar
  17. 17.
    M. G. Krein and M. A. Krasnosel'skii, “Main theorems on extensions of Hermitian operators and some of their applications to the theory of orthogonal polynomials and the moment problem,”Usp. Mat. Nauk,2, Issue 3, 60–106 (1947).Google Scholar
  18. 18.
    M. G. Krein, “InfiniteJ-matrices and the matrix moment problem,”Dokl. Akad. Nauk SSSR,69, 125–128 (1949).Google Scholar
  19. 19.
    Yu. M. Berezanskii,Expansions in Eigenfunctions of Self-Adjoint Operators, American Mathematical Society, Providence, RI (1965).Google Scholar
  20. 20.
    M. G. Krein, “An analog of the Chebyshev-Markov inequalities in a one-dimensional boundary-value problem,”Dokl. Akad. Nauk SSSR,89, Nos. 1, 5–8 (1953).Google Scholar
  21. 21.
    M. G. Krein, “Chebyshev-Markov inequalities in the theory of spectral functions of strings,”Mat. Issled.,5, No. 1, 77–101 (1979).Google Scholar
  22. 22.
    M. G. Krein, “On the description of all solutions of the truncated power moment problem and some operator problems,”Mat. Issled.,2, No. 2, 114–132 (1968).Google Scholar
  23. 23.
    M. G. Krein and H. Langer, “On the indefinite power moment problem,”Dokl. Akad. Nauk SSSR,226, No. 2, 261–264 (1976).Google Scholar
  24. 24.
    M. G. Krein and H. Langer, “On some extension problems which are closely connected with the theory of Hermitian operators in a space ΠK. III,” in:Indefinite Analogues of the Hamburger and Stieltjes moment problems. Pt 1. Beitrage zur Analysis, Vol. 14, (1979), pp. 25–40.Google Scholar
  25. 25.
    V. A. Fil'shtinskii, “Power moment problem on the entire axis for a given finite number of empty intervals in the spectrum,”Zapiski Mekh.-Mat. Fakult. Khark. Univ., Khark. Mat. Obshch., Ser. 4,30, 186–200 (1964).Google Scholar
  26. 26.
    A. A. Nudel'man, “Canonical solutions of the moment problem on several intervals,”Mat. Zametki,1, No. 4, 435–442 (1967).Google Scholar
  27. 27.
    V. A. Derkach and M. M. Malamud, “Resolvent matrices of Hermitian operators and the moment problem with gaps,”Dokl. Akad. Nauk SSSR,314, No. 2, 273–278 (1990).Google Scholar
  28. 28.
    T. Stieltjes,Studies on Continued Fractions [Russian translation], ONTI, Moscow (1936).Google Scholar
  29. 29.
    A. A. Markov, “Lectures on functions with the least deviations from zero,” in:Selected Works [in Russian], Gostekhizdat, Moscow (1948), pp. 244–291.Google Scholar
  30. 30.
    N. I. Akhiezer, “On the Lamé equation,”Zapiski Mekh.-Mat. Fakult. Khark. Univ., Khark. Mat. Obshch., Ser. 4,30, 4–57 (1962).Google Scholar
  31. 31.
    M. G. Krein, B. Ya. Levin, and A. A. Nudel'man,On Special Representations of Polynomials Positive on a System of Closed Intervals [in Russian], Preprint 28-84, Physicotechnical Institute of Low Temperatures, Ukrainian Academy of Sciences, Kharkov (1984).Google Scholar
  32. 32.
    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, University Press, Oxford (1990).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • A. A. Nudel'man
    • 1
  1. 1.Odessa Construction Engineering instituteOdessa

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