Existence of smooth solutions to the classical moment problems
Author:
Palle E. T. Jorgensen
Journal:
Trans. Amer. Math. Soc. 332 (1992), 839848
MSC:
Primary 44A60; Secondary 42A70, 43A35, 46N99, 47A57
MathSciNet review:
1059709
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a given sequence, and define for . If holds for all finite sequences , then it is known that there is a positive Borel measure on the circle such that , and conversely. Our main theorem provides a necessary and sufficient condition on the sequence that the measure may be chosen to be smooth. A measure is said to be smooth if it has the same spectral type as the operator acting on with respect to Haar measure on : Equivalently, is a superposition (possibly infinite) of measures of the form with such that . The condition is stated purely in terms of the initially given sequence : We show that a smooth representation exists if and only if, for some , the a priori estimate is valid for all finite double sequences . 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.
 [Ak]
N. I. Akhiezer, The classical moment problem, Oliver and Boyd, Edinburgh and London, 1965.
 [AG]
N. I. Akhiezer and I. M. Glazman, The theory of linear operators in Hilbert space, vol. II, (2nd ed.), Ungar, New York, 1966.
 [BC]
Ch.
Berg and J.
P. R. Christensen, Density questions in the classical theory of
moments, Ann. Inst. Fourier (Grenoble) 31 (1981),
no. 3, vi, 99–114 (English, with French summary). MR 638619
(84i:44006)
 [BR]
Ola
Bratteli and Derek
W. Robinson, Operator algebras and quantum statistical mechanics.
1, 2nd ed., Texts and Monographs in Physics, SpringerVerlag, New
York, 1987. 𝐶* and 𝑊*algebras, symmetry groups,
decomposition of states. MR 887100
(88d:46105)
 [DS]
N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1967.
 [Ha]
Hans
Ludwig Hamburger, Hermitian transformations of deficiencyindex
(1,1), Jacobi matrices and undetermined moment problems, Amer. J.
Math. 66 (1944), 489–522. MR 0011169
(6,130d)
 [Jo1]
Palle
E. T. Jørgensen, A uniqueness theorem for the
HeisenbergWeyl commutation relations with nonselfadjoint position
operator, Amer. J. Math. 103 (1981), no. 2,
273–287. MR
610477 (82g:81033), http://dx.doi.org/10.2307/2374217
 [Jo2]
Palle
E. T. Jorgensen, Operators and representation theory,
NorthHolland Mathematics Studies, vol. 147, NorthHolland Publishing
Co., Amsterdam, 1988. Canonical models for algebras of operators arising in
quantum mechanics; Notas de Matemática [Mathematical Notes], 120. MR 919948
(89e:47001)
 [JMo]
Palle
E. T. Jorgensen and Robert
T. Moore, Operator commutation relations, Mathematics and its
Applications, D. Reidel Publishing Co., Dordrecht, 1984. Commutation
relations for operators, semigroups, and resolvents with applications to
mathematical physics and representations of Lie groups. MR 746138
(86i:22006)
 [JMu]
Palle
T. Jørgensen and Paul
S. Muhly, Selfadjoint extensions satisfying the Weyl operator
commutation relations, J. Analyse Math. 37 (1980),
46–99. MR
583632 (82k:47058), http://dx.doi.org/10.1007/BF02797680
 [JPo]
P. E. T. Jorgensen and R. T. Powers, Positive elements for the quantum problem of moments, Preprint, 1989.
 [La1]
H.
J. Landau, Classical background of the moment problem, Moments
in mathematics (San Antonio, Tex., 1987) Proc. Sympos. Appl. Math.,
vol. 37, Amer. Math. Soc., Providence, RI, 1987, pp. 1–15.
MR 921082
(89g:44001)
 [La2]
H.
J. Landau, Maximum entropy and the moment
problem, Bull. Amer. Math. Soc. (N.S.)
16 (1987), no. 1,
47–77. MR
866018 (88k:42010), http://dx.doi.org/10.1090/S027309791987154644
 [Po]
Robert
T. Powers, Selfadjoint algebras of unbounded
operators. II, Trans. Amer. Math. Soc. 187 (1974), 261–293.
MR
0333743 (48 #12067), http://dx.doi.org/10.1090/S00029947197403337438
 [We]
Reinhard
F. Werner, Dilations of symmetric operators shifted by a unitary
group, J. Funct. Anal. 92 (1990), no. 1,
166–176. MR 1064692
(92e:47068), http://dx.doi.org/10.1016/00221236(90)90073T
 [Ak]
 N. I. Akhiezer, The classical moment problem, Oliver and Boyd, Edinburgh and London, 1965.
 [AG]
 N. I. Akhiezer and I. M. Glazman, The theory of linear operators in Hilbert space, vol. II, (2nd ed.), Ungar, New York, 1966.
 [BC]
 Ch. Berg and J. P. R. Christensen, Density questions in the classical theory of moments, Ann. Inst. Fourier (Grenoble) 31 (1981), 99114. MR 638619 (84i:44006)
 [BR]
 O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics, vol. I (2nd ed.), SpringerVerlag, New York, 1987. MR 887100 (88d:46105)
 [DS]
 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), 489522. MR 0011169 (6:130d)
 [Jo1]
 P. E. T. Jorgensen, A uniqueness theorem for the HeisenbergWeyl commutation relations with nonselfadjoint position operator, Amer. J. Math. 103 (1980), 273287. MR 610477 (82g:81033)
 [Jo2]
 , Operators and representation theory, NorthHolland, Amsterdam, 1988. MR 919948 (89e:47001)
 [JMo]
 P. E. T. Jorgensen and R. T. Moore, Operator commutation relations, Reidel, Dordrecht and Boston, Mass., 1984. MR 746138 (86i:22006)
 [JMu]
 P. E. T. Jorgensen and P. S. Muhly, Selfadjoint extensions satisfying the Weyl operator commutation relations, J. Analyse Math. 37 (1980), 4699. MR 583632 (82k:47058)
 [JPo]
 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), 4777. MR 866018 (88k:42010)
 [Po]
 R. T. Powers, Selfadjoint algebras of unbounded operators. II, Trans. Amer. Math. Soc. 187 (1974), 261293. MR 0333743 (48:12067)
 [We]
 R. F. Werner, Dilations of symmetric operators shifted by a unitary group, Preprint 1989, J. Funct. Anal. 92 (1990), 166176. 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:
http://dx.doi.org/10.1090/S00029947199210597091
PII:
S 00029947(1992)10597091
Article copyright:
© Copyright 1992 American Mathematical Society
