Density questions in the classical theory of moments
Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 99-114.

Soit μ une mesure de Radon positive sur la droite dont tous les moments existent. Nous démontrons que l’ensemble P des polynômes n’est pas dense dans L p (R,μ) pour p>2, si μ est indéterminée. Si μ est déterminée P est dense dans L p (R,μ) pour 1p2, mais non nécessairement pour p>2. Ensuite, nous étudions l’ensemble convexe et compact des mesures de Radon positives admettant les mêmes moments que μ.

Let μ be a positive Radon measure on the real line having moments of all orders. We prove that the set P of polynomials is note dense in L p (R,μ) for any p>2, if μ is indeterminate. If μ is determinate, then P is dense in L p (R,μ) for 1p2, but not necessarily for p>2. The compact convex set of positive Radon measures with same moments as μ is studied in some details.

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     title = {Density questions in the classical theory of moments},
     journal = {Annales de l'Institut Fourier},
     pages = {99--114},
     publisher = {Institut Fourier},
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     volume = {31},
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Berg, Christian; Christensen, J. P. Reus. Density questions in the classical theory of moments. Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 99-114. doi : 10.5802/aif.840. http://www.numdam.org/articles/10.5802/aif.840/

[1] N.I. Akhiezer, The classical moment problem, Oliver and Boyd, Edinburgh, 1965. | Zbl

[2] H. Bauer, Wahrscheinlichkeitstheorie und Grundzüge der Masstheorie, De Gruyter, Berlin, 1978. | Zbl

[3] G. Freud, Orthogonal polynomials, Pergamon Press, Oxford, 1971. | Zbl

[4] E. Hewitt, Remark on orthonormal sets in L2 (a, b), Amer. Math. Monthly, 61 (1954), 249-250. | MR | Zbl

[5] M.A. Naimark, Extremal spectral functions of a symmetric operator, Izv. Akad. Nauk. SSSR, ser. matem., 11 ; Dokl. Akad. Nauk. SSSR, 54 (1946), 7-9. | MR | Zbl

[6] R.R. Phelps, Lectures on Choquet's Theorem, Van Nostrand, New York, 1966. | MR | Zbl

[7] M. Riesz, Sur le problème des moments et le théorème de Parseval correspondant, Acta Litt. Ac. Sci., Szeged., 1 (1923), 209-225. | JFM

[8] J.A. Shohat and J.D. Tamarkin, The problem of moments, AMS, New York, 1943. | MR | Zbl

[9] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean spaces, Princeton University Press, 1971. | MR | Zbl

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