Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Existence of smooth solutions to the classical moment problems


Author: Palle E. T. Jorgensen
Journal: Trans. Amer. Math. Soc. 332 (1992), 839-848
MSC: Primary 44A60; Secondary 42A70, 43A35, 46N99, 47A57
DOI: https://doi.org/10.1090/S0002-9947-1992-1059709-1
MathSciNet review: 1059709
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ s(0),s(1), \ldots $ be a given sequence, and define $ s(n) = \overline {s( - n)} $ for $ n < 0$. If $ \Sigma \Sigma {\overline \xi _n}{\xi _m}s(m - n) \geq 0$ holds for all finite sequences $ {({\xi _n})_{n \in \mathbb{Z}}}$, then it is known that there is a positive Borel measure $ \mu $ on the circle $ \mathbb{T}$ such that $ s(n) = \smallint_{ - \pi }^\pi {{e^{int}}d\mu (t)} $, and conversely. Our main theorem provides a necessary and sufficient condition on the sequence $ (s(n))$ that the measure $ \mu $ may be chosen to be smooth. A measure $ \mu $ is said to be smooth if it has the same spectral type as the operator $ id/dt$ acting on $ {L^2}(\mathbb{T})$ with respect to Haar measure $ dt$ on $ \mathbb{T}$: Equivalently, $ \mu $ is a superposition (possibly infinite) of measures of the form $ \vert w(t){\vert^2}dt$ with $ w \in {L^2}(\mathbb{T})$ such that $ dw/dt \in {L^2}(\mathbb{T})$. The condition is stated purely in terms of the initially given sequence $ (s(n))$: We show that a smooth representation exists if and only if, for some $ \varepsilon \in {\mathbb{R}_ + }$, the a priori estimate

$\displaystyle \sum {\sum {s(m - n){{\overline \xi }_n}{\xi _m} \geq \varepsilon {{\left\vert {\sum {ns(n){\xi _n}} } \right\vert}^2}} } $

is valid for all finite double sequences $ ({\xi _n})$. An analogous result is proved for the determinate (Hamburger) moment problem on the line. But the corresponding result does not hold for the indeterminate moment problem.

References [Enhancements On Off] (What's this?)

  • [Ak] N. I. Akhiezer, The classical moment problem, Oliver and Boyd, Edinburgh and London, 1965.
  • [A-G] N. I. Akhiezer and I. M. Glazman, The theory of linear operators in Hilbert space, vol. II, (2nd ed.), Ungar, New York, 1966.
  • [B-C] Ch. Berg and J. P. R. Christensen, Density questions in the classical theory of moments, Ann. Inst. Fourier (Grenoble) 31 (1981), 99-114. MR 638619 (84i:44006)
  • [B-R] O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics, vol. I (2nd ed.), Springer-Verlag, New York, 1987. MR 887100 (88d:46105)
  • [D-S] N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1967.
  • [Ha] H. Hamburger, Hermitian transformations of deficiency matrix (1.1), Jacobi matrices, and undetermined moment problems, Amer. J. Math. 66 (1944), 489-522. MR 0011169 (6:130d)
  • [Jo1] P. E. T. Jorgensen, A uniqueness theorem for the Heisenberg-Weyl commutation relations with non-selfadjoint position operator, Amer. J. Math. 103 (1980), 273-287. MR 610477 (82g:81033)
  • [Jo2] -, Operators and representation theory, North-Holland, Amsterdam, 1988. MR 919948 (89e:47001)
  • [J-Mo] P. E. T. Jorgensen and R. T. Moore, Operator commutation relations, Reidel, Dordrecht and Boston, Mass., 1984. MR 746138 (86i:22006)
  • [J-Mu] P. E. T. Jorgensen and P. S. Muhly, Self-adjoint extensions satisfying the Weyl operator commutation relations, J. Analyse Math. 37 (1980), 46-99. MR 583632 (82k:47058)
  • [J-Po] P. E. T. Jorgensen and R. T. Powers, Positive elements for the quantum problem of moments, Preprint, 1989.
  • [La1] H. J. Landau (ed.), Moments in mathematics, Proc. Sympos. Appl. Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1987. MR 921082 (89g:44001)
  • [La2] -, Maximum entropy and the moment problem, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 47-77. MR 866018 (88k:42010)
  • [Po] R. T. Powers, Selfadjoint algebras of unbounded operators. II, Trans. Amer. Math. Soc. 187 (1974), 261-293. MR 0333743 (48:12067)
  • [We] R. F. Werner, Dilations of symmetric operators shifted by a unitary group, Preprint 1989, J. Funct. Anal. 92 (1990), 166-176. MR 1064692 (92e:47068)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 44A60, 42A70, 43A35, 46N99, 47A57

Retrieve articles in all journals with MSC: 44A60, 42A70, 43A35, 46N99, 47A57


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1059709-1
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society