On M. G. Krein's contribution to the moment problem
- A. A. Nudel'man
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This is a survey of M. G. Krein' s papers devoted to the moment problem.
Translated from Ukrainskii Matematicheskii Zhumal, Vol. 46, Nos. 1–2, pp. 63–75, January–February, 1994.
Henry Landau (editor),Moments in Mathematics, Proceedings of Symposia in Applied Mathematics, Vol. 37, American Mathematical Society, Providence, RI (1987).
N. I. Akhiezer and M. G. Krein,On Some Problems in the Theory of Moments [in Russian], GONTI, Kharkov (1938).
A. A. Markov,New Applications of Continued Fractions. Selected Works [in Russian], Gostekhizdat, Moscow (1948).
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).
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).
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.
M. G. Krein, “On positive additive functional in linear normed spaces,”Zapiski Mat. Obshch.,4, No. 14, 227–237 (1937).
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).
S. Karlin and W. Studden,Tchebycheff Systems: with Application in Analysis and Statistics, Interscience (1966).
M. G. Krein and A. A. Nudel'man,Markov Moment Problem and Extremal Problems [in Russian], Nauka, Moscow (1973).
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).
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).
I. G. Kaplan,Introduction to the Theory of Intermolecular Interactions [in Russian], Nauka, Moscow (1982).
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).
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).
N. I. Akhiezer and M. G. Krein. “An extremum problem for polynomials,”Ann. Univ. Sci. Budapest, Sec. Math.,3/4, 9–14 (1960/61).
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).
M. G. Krein, “InfiniteJ-matrices and the matrix moment problem,”Dokl. Akad. Nauk SSSR,69, 125–128 (1949).
Yu. M. Berezanskii,Expansions in Eigenfunctions of Self-Adjoint Operators, American Mathematical Society, Providence, RI (1965).
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).
M. G. Krein, “Chebyshev-Markov inequalities in the theory of spectral functions of strings,”Mat. Issled.,5, No. 1, 77–101 (1979).
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).
M. G. Krein and H. Langer, “On the indefinite power moment problem,”Dokl. Akad. Nauk SSSR,226, No. 2, 261–264 (1976).
M. G. Krein and H. Langer, “On some extension problems which are closely connected with the theory of Hermitian operators in a space Π K . III,” in:Indefinite Analogues of the Hamburger and Stieltjes moment problems. Pt 1. Beitrage zur Analysis, Vol. 14, (1979), pp. 25–40.
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).
A. A. Nudel'man, “Canonical solutions of the moment problem on several intervals,”Mat. Zametki,1, No. 4, 435–442 (1967).
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).
T. Stieltjes,Studies on Continued Fractions [Russian translation], ONTI, Moscow (1936).
A. A. Markov, “Lectures on functions with the least deviations from zero,” in:Selected Works [in Russian], Gostekhizdat, Moscow (1948), pp. 244–291.
N. I. Akhiezer, “On the Lamé equation,”Zapiski Mekh.-Mat. Fakult. Khark. Univ., Khark. Mat. Obshch., Ser. 4,30, 4–57 (1962).
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).
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).
- On M. G. Krein's contribution to the moment problem
Ukrainian Mathematical Journal
Volume 46, Issue 1-2 , pp 62-74
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- A. A. Nudel'man (1)
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- 1. Odessa Construction Engineering institute, Odessa