Abstract
In Chap. 16, we present a short and concise treatment of the one-dimensional Hamburger moment problem with an emphasis on the self-adjoint extension theory. Orthogonal polynomials and the Jacobi operator associated with a moment sequence are developed. Basic results on the existence, the set of solutions, and the uniqueness of the moment problem are given in terms of self-adjoint extensions. The two final sections of this chapter are devoted to the advanced theory of the indeterminate case. The four Nevanlinna functions and the Weyl circles are defined and studied, and all von Neumann solutions are described. This chapter ends with a proof of Nevanlinna’s fundamental theorem on the parameterization of the set of all solutions of the indeterminate moment problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Classical Articles
Calkin, J.W.: Abstract symmetric boundary conditions. Trans. Am. Math. Soc. 45, 369–442 (1939)
Hamburger, H.L.: Über eine Erweiterung des Stieltjesschen Momentenproblems. Math. Ann. 81, 235–319 (1920) and 82, 120–164, 168–187 (1920)
Krein, M.G.: On Hermitian operators with defect numbers one. Dokl. Akad. Nauk SSSR 43, 339–342 (1944)
Krein, M.G.: On resolvents of an Hermitian operator with defect index (m,m). Dokl. Akad. Nauk SSSR 52, 657–660 (1946)
Krein, M.G.: The theory of self-adjoint extensions of semibounded Hermitian operators and its applications. Mat. Sb. 20, 365–404 (1947)
Naimark, M.A.: Spectral functions of a symmetric operator. Izv. Akad. Nauk SSSR, Ser. Mat. 7, 285–296 (1943)
Nevanlinna, R.: Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjessche Momentenproblem. Ann. Acad. Sci. Fenn. A 18, 1–53 (1922)
Stieltjes, T.J.: Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8, 1–122 (1894)
Von Neumann, J.: Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Math. Ann. 102, 49–131 (1929)
Weyl, H.: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68, 220–269 (1910)
Books
Akhiezer, N.I.: The Classical Moment Problem. Oliver and Boyd, Edinburgh and London (1965)
Amrein, W.O., Hinz, A.M., Pearson, D.B. (eds.): Sturm–Liouville Theory. Past and Present. Birkhäuser-Verlag, Basel (2005)
Berezansky, Y.M.: Expansions in Eigenfunctions of Selfadjoint Operators. Am. Math. Soc., Providence (1968)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer-Verlag, Berlin (2011)
Dunford, N., Schwartz, J.T.: Linear Operators, Part II. Spectral Theory. Interscience Publ., New York (1963)
Gorbachuk, V.I., Gorbachuk, M.L.: Boundary Value Problems for Operator Differential Equations. Kluwer, Dordrecht (1991)
Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer-Verlag, Berlin (1965)
Jörgens, K., Rellich, F.: Eigenwerttheorie Gewöhnlicher Differentialgleichungen. Springer-Verlag, Berlin (1976)
Naimark, M.A.: Linear Differential Operators. Ungar, New York (1968)
Shohat, J.A., Tamarkin, J.D.: The Problem of Moments. Am. Math. Soc., Providence (1943)
Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I. Clarendon Press, Oxford (1962). Part II (1970)
Articles
Alonso, A., Simon, B.: The Birman–Krein–Vishik theory of selfadjoint extensions of semibounded operators. J. Oper. Theory 4, 51–270 (1980)
Ando, T., Nishio, K.: Positive selfadjoint extensions of positive symmetric operators. Tohoku Math. J. 22, 65–75 (1970)
Arens, R.: Operational calculus of linear relations. Pac. J. Math. 11, 9–23 (1961)
Arlinski, Y.M., Hassi, S., Sebestyen, Z., de Snoo, H.S.V.: On the class of extremal extensions of a nonnegative operator. Oper. Theory Adv. Appl. 127, 41–81 (2001)
Arlinski, Y., Tsekanovskii, E.: The von Neumann problem for nonnegative symmetric operators. Integral Equ. Oper. Theory 51, 319–356 (2005)
Bennewitz, C., Everitt, W.N.: The Weyl–Titchmarsh eigenfunction expansion theorem for Sturm–Liouville operators. In [AHP], pp. 137–172
Birman, M.S.: On the theory of selfadjoint extensions of positive operators. Mat. Sb. 38, 431–450 (1956)
Brasche, J.F., Malamud, M.M., Neidhardt, N.: Weyl function and spectral properties of selfadjoint extensions. Integral Equ. Oper. Theory 43, 264–289 (2002)
Brown, B.M., Grubb, G., Wood, I.: M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. Math. Nachr. 282, 314–347 (2009)
Bruk, V.M.: On a class of boundary value problems with a spectral parameter in the boundary condition. Mat. Sb. 100, 210–216 (1976)
Brüning, J., Geyler, V., Pankrashkin, K.: Spectra of selfadjoint extensions and applications to solvable Schrödinger operators. Rev. Math. Phys. 20, 1–70 (2008)
Buchwalther, H., Casier, G.: La paramétrisation de Nevanlinna dans le problème des moments de Hamburger. Expo. Math. 2, 155–178 (1984)
Coddington, E.A.: Extension theory of formally normal and symmetric subspaces. Mem. Am. Math. Soc. 134 (1973)
Coddington, E.A., de Snoo, H.S.V.: Positive selfadjoint extensions of positive subspaces. Math. Z. 159, 203–214 (1978)
Derkach, V.A., Malamud, M.M.: Generalized resolvents and the boundary value problem for Hermitian operators with gaps. J. Funct. Anal. 95, 1–95 (1991)
Dijksma, A., de Snoo, H.S.V.: Selfadjoint extensions of symmetric subspaces. Pac. J. Math. 54, 71–100 (1974)
Grampp, U., Wähnert, Ph.: Das Hamburger’sche Momentenproblem und die Nevanlinna-Parametrisierung. Diplomarbeit, Leipzig (2009)
Hassi, S., Malamud, M., de Snoo, H.S.V.: On Krein’s extension theory of nonnegative operators. Math. Nachr. 274, 40–73 (2004)
Kochubei, A.N.: Extensions of symmetric operators. Math. Notes 17, 25–28 (1975)
Kodaira, K.: Eigenvalue problems for ordinary differential equations of the second order and Heisenberg’s theory of S-matrices. Am. J. Math. 71, 921–945 (1950)
Kodaira, K.: On ordinary differential equations of any even order and the corresponding eigenfunction expansions. Am. J. Math. 72, 502–544 (1950)
Malamud, M.M.: Certain classes of extensions of a lacunary Hermitian operator. Ukr. Mat. Zh. 44, 215–233 (1992)
Malamud, M.M., Neidhardt, H.: On the unitary equivalence of absolutely continuous parts of self-adjoint extensions. J. Funct. Anal. 260, 613–638 (2011)
Posilicano, A.: A Krein-type formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal. 183, 109–147 (2001)
Prokaj, V., Sebestyen, Z.: On extremal positive extensions. Acta Sci. Math. Szeged 62, 485–491 (1996)
Saakyan, S.N.: Theory of resolvents of symmetric operators with infinite deficiency indices. Dokl. Akad. Nauk Arm. SSR 41, 193–198 (1965)
Sebestyen, Z., Stochel, J.: Restrictions of positive selfadjoint operators. Acta Sci. Math. (Szeged) 55, 149–154 (1991)
Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137, 82–203 (1998)
Vishik, M.: On general boundary conditions for elliptic differential equations. Tr. Moskv. Mat. Obc. 1, 187–246 (1952)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Schmüdgen, K. (2012). The One-Dimensional Hamburger Moment Problem. In: Unbounded Self-adjoint Operators on Hilbert Space. Graduate Texts in Mathematics, vol 265. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4753-1_16
Download citation
DOI: https://doi.org/10.1007/978-94-007-4753-1_16
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-4752-4
Online ISBN: 978-94-007-4753-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)