The classical moment problem as a self-adjoint finite difference operator. (English) Zbl 0910.44004

In this comprehensive exposition, the author discusses the classical moment problem from the theory of finite difference operators. The Stieltjes and the Hamburger moment problems are considered from the self-adjointness point of view. As an advantage of this approach it is shown that the Nevanlinna functions appear as elements of a transfer matrix and the convergence of Padé approximants is a strong resolvent convergence of finite matrix approximations to a Jacobi matrix. New results on the convergence of certain Padé approximants for Hamburger series are obtained.


44A60 Moment problems
47A57 Linear operator methods in interpolation, moment and extension problems
39A70 Difference operators
41A21 Padé approximation
Full Text: DOI arXiv Link


[1] Akhiezer, N. I., The Classical Moment Problem (1965), Hafner: Hafner New York · Zbl 0135.33803
[2] Akhiezer, N. I.; Glazman, I. M., Theory of Linear Operators in Hilbert Space (1961), Ungar: Ungar New York · Zbl 0098.30702
[3] Alonso, A.; Simon, B., The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators, J. Operator Theory, 4, 251-270 (1980) · Zbl 0467.47017
[4] Baker, G.; Graves-Morris, P., Padé Approximants (1996), Cambridge Univ. Press: Cambridge Univ. Press New York
[5] Dunford, N.; Schwartz, J., Linear Operators, II. Spectral Theory (1963), Interscience: Interscience New York
[7] de Lamadrid, J. Gil, Determinacy theory for the Livšic moments problem, J. Math. Anal. Appl., 34, 429-444 (1971) · Zbl 0224.47012
[8] Hamburger, H., Über eine Erweiterung des Stieltjesschen Momentproblems, Math. Ann., 81, 235-319 (1920) · JFM 47.0427.04
[9] Ismail, M. E.H.; Masson, D., \(qq\), Trans. Amer. Math. Soc., 346, 63-116 (1994)
[12] Kato, T., Perturbation Theory of Linear Operators (1980), Springer-Verlag: Springer-Verlag New York
[13] Katznelson, Y., An Introduction to Harmonic Analysis (1976), Dover: Dover New York · Zbl 0169.17902
[15] Krein, M. G., Chebyshev’s and Markov’s ideas in the theory of limiting values of integrals and their further development, Uspekhi Matem. Nauk., 44 (1951)
[16] Krein, M. G., On a generalization of an investigation by Stieltjes, Dokl. Akad. Nauk SSSR, 87 (1952) · Zbl 0049.34702
[17] Krein, M. G., Some new problems in the theory of oscillations of Sturm systems, Prikl. Mat. Mekh., 16, 555-568 (1952)
[18] Krein, M. G., On an extrapolation problem due to Kolmogorov, Dokl. Akad. Nauk. SSSR, 46, 306-309 (1945) · Zbl 0063.03356
[19] Landau, H., The classical moment problem: Hilbertian proofs, J. Funct. Anal., 38, 255-272 (1980) · Zbl 0446.44006
[20] Langer, R. W., More determinacy theory for the Livšic moments problem, J. Math. Anal. Appl., 56, 586-616 (1976) · Zbl 0353.44009
[21] Livšic, M. S., On an application of the theory of hermitian operators to the generalized problem of moments, Dokl. Akad. Nauk. SSSR, 26, 17-22 (1940)
[22] Loeffel, J. J.; Martin, A.; Simon, B.; Wightman, A. S., Padé approximants and the anharmonic oscillator, Phys. Lett. B, 30, 656-658 (1969)
[23] Masson, D.; McClary, W., Classes of \(C^∞\), J. Funct. Anal., 10, 19-32 (1972) · Zbl 0234.47026
[24] Naimark, M. A., Self-adjoint extensions of the second kind of a symmetric operator, Izv. Akad. Nauk SSSR, 4, 90-104 (1940) · Zbl 0025.06402
[25] Naimark, M. A., Spectral functions of a symmetric operator, Izv. Akad. Nauk SSSR, 4, 309-318 (1940) · Zbl 0025.06403
[26] Naimark, M. A., On spectral functions of a symmetric operator, Izv. Akad. Nauk SSSR, 7, 285-296 (1943) · Zbl 0061.26005
[27] Naimark, M. A., On extremal spectral functions of a symmetric operator, Dokl. Akad. Nauk SSSR, 54, 7-9 (1946) · Zbl 0061.26006
[28] Nelson, E., Analytic vectors, Ann. of Math., 70, 572-615 (1959) · Zbl 0091.10704
[29] Nevanlinna, R., Asymptotische Entwickelungen beschränkter Functionen und das Stieltjessche Momentenproblem, Ann. Acad. Sci. Fenn. A, 18 (1922) · JFM 48.1226.02
[30] Nussbaum, A., Quasi-analytic vectors, Ark. Mat., 6, 179-191 (1965) · Zbl 0182.46102
[31] Nussbaum, A., A note on quasi-analytic vectors, Studia Math., 33, 305-310 (1969) · Zbl 0189.43903
[32] Pearson, D. B., Quantum Scattering and Spectral Theory (1988), Academic Press: Academic Press London · Zbl 0673.47011
[33] Reed, M.; Simon, B., Methods of Modern Mathematical Physics, I. Functional Analysis (1972), Academic Press: Academic Press New York
[34] Reed, M.; Simon, B., Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness (1975), Academic Press: Academic Press New York · Zbl 0308.47002
[35] Shohat, J. A.; Tamarkin, J. D., The problem of moments, Amer. Math. Soc. Surveys, 1 (1943) · Zbl 0063.06973
[36] Simon, B., Coupling constant analyticity for the anharmonic oscillator, Ann. Phys., 58, 76-136 (1970)
[37] Simon, B., A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal., 28, 377-385 (1978) · Zbl 0413.47029
[38] Stieltjes, T., Recherches sur les fractions continues, Anns. Fac. Sci. Univ. Toulouse, 8, J1-J122 (1894-1895) · JFM 25.0326.01
[39] Stone, M. H., Linear Transformations in Hilbert Space. Linear Transformations in Hilbert Space, Amer. Math. Soc. Colloq. (1932), Publ. XV: Publ. XV New York · Zbl 0005.16403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.