Moment problems for compact sets

Chandler, J. D.

doi:10.1090/S0002-9939-1988-0942632-2

Moment problems for compact sets
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by J. D. Chandler

Proc. Amer. Math. Soc. 104 (1988), 1134-1140

DOI: https://doi.org/10.1090/S0002-9939-1988-0942632-2

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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.

References

N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR 0184042
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, DOI 10.1090/psapm/037/921086
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, DOI 10.1007/BF01420423
Christian Berg and P. H. Maserick, Polynomially positive definite sequences, Math. Ann. 259 (1982), no. 4, 487–495. MR 660043, DOI 10.1007/BF01466054
Gilles Cassier, Problème des moments sur un compact de $\textbf {R}^{n}$ et décomposition de polynômes à plusieurs variables, J. Funct. Anal. 58 (1984), no. 3, 254–266 (French). MR 759099, DOI 10.1016/0022-1236(84)90042-9
William F. Donoghue Jr., Monotone matrix functions and analytic continuation, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR 0486556
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
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
Felix Hausdorff, Summationsmethoden und Momentfolgen. I, Math. Z. 9 (1921), no. 1-2, 74–109 (German). MR 1544453, DOI 10.1007/BF01378337
Einar Hille, Introduction to general theory of reproducing kernels, Rocky Mountain J. Math. 2 (1972), no. 3, 321–368. MR 315109, DOI 10.1216/RMJ-1972-2-3-321
M. G. Kreĭn and A. A. Nudel′man, The Markov moment problem and extremal problems, Translations of Mathematical Monographs, Vol. 50, 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. MR 0458081
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, DOI 10.1090/coll/015
M. H. Stone, The generalized Weierstrass approximation theorem, Math. Mag. 21 (1948), 167–184, 237–254. MR 27121, DOI 10.2307/3029750
Béla Sz.-Nagy, Spektraldarstellung linearer Transformationen des Hilbertschen Raumes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 39, Springer-Verlag, Berlin-New York, 1967 (German). Berichtigter Nachdruck. MR 0213890