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Moment problems for compact sets


Author: J. D. Chandler
Journal: Proc. Amer. Math. Soc. 104 (1988), 1134-1140
MSC: Primary 44A60; Secondary 42A70, 47B15
MathSciNet review: 942632
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Abstract: The solvability of the Hausdorff moment problem for an arbitrary compact subset of Euclidean $ n$-space is shown to be equivalent to the nonnegativity of a family of quadratic forms derived from the given moment sequence and the given compact set. A variant theorem for the one-dimensional case and an analogous theorem for the trigonometric moment problem are also given. The one-dimensional theorems are similar to theorems of Kreĭn and Nudel'man [11], but the proofs, unlike those in [11], do not depend on the existence of a standard form for polynomials which are nonnegative on a given compact set.


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  • [1] N. I. Akhiezer, The classical moment problem and some related questions in analysis, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965. MR 0184042
  • [2] Christian Berg, The multidimensional moment problem and semigroups, Moments in mathematics (San Antonio, Tex., 1987) Proc. Sympos. Appl. Math., vol. 37, Amer. Math. Soc., Providence, RI, 1987, pp. 110–124. MR 921086, 10.1090/psapm/037/921086
  • [3] C. Berg, J. P. R. Christensen, and C. U. Jensen, A remark on the multidimensional moment problem, Math. Ann. 243 (1979), no. 2, 163–169. MR 543726, 10.1007/BF01420423
  • [4] Christian Berg and P. H. Maserick, Polynomially positive definite sequences, Math. Ann. 259 (1982), no. 4, 487–495. MR 660043, 10.1007/BF01466054
  • [5] Gilles Cassier, Problème des moments sur un compact de 𝑅ⁿ et décomposition de polynômes à plusieurs variables, J. Funct. Anal. 58 (1984), no. 3, 254–266 (French). MR 759099, 10.1016/0022-1236(84)90042-9
  • [6] William F. Donoghue Jr., Monotone matrix functions and analytic continuation, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 207. MR 0486556
  • [7] V. A. Fil′štinskiĭ, The power moment problem on the entire axis with a given finite number of empty intervals in the spectrum, Zap. Meh.-Mat. Fak. Har′kov. Gos. Univ. i Har′kov. Mat. Obšč. (4) 30 (1964), 186–200 (Russian). MR 0209780
  • [8] Paul R. Halmos, Introduction to Hilbert space and the theory of spectral multiplicity, AMS Chelsea Publishing, Providence, RI, 1998. Reprint of the second (1957) edition. MR 1653399
  • [9] Felix Hausdorff, Summationsmethoden und Momentfolgen. I, Math. Z. 9 (1921), no. 1-2, 74–109 (German). MR 1544453, 10.1007/BF01378337
  • [10] Einar Hille, Introduction to general theory of reproducing kernels, Rocky Mountain J. Math. 2 (1972), no. 3, 321–368. MR 0315109
  • [11] M. G. Kreĭn and A. A. Nudel′man, The Markov moment problem and extremal problems, American Mathematical Society, Providence, R.I., 1977. Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development; Translated from the Russian by D. Louvish; Translations of Mathematical Monographs, Vol. 50. MR 0458081
  • [12] Marshall Harvey Stone, Linear transformations in Hilbert space, American Mathematical Society Colloquium Publications, vol. 15, American Mathematical Society, Providence, RI, 1990. Reprint of the 1932 original. MR 1451877
  • [13] M. H. Stone, The generalized Weierstrass approximation theorem, Math. Mag. 21 (1948), 167–184, 237–254. MR 0027121
  • [14] Béla Sz.-Nagy, Spektraldarstellung linearer Transformationen des Hilbertschen Raumes, Berichtigter Nachdruck. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 39, Springer-Verlag, Berlin-New York, 1967 (German). MR 0213890

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0942632-2
Article copyright: © Copyright 1988 American Mathematical Society