×

Rotation invariant moment problems. (English) Zbl 0744.44006

A famous theorem of M. Riesz states that the polynomials in one variable are dense in \(L^ 2(\mu)\) when \(\mu\) is a determinate measure on the real line. It has not been known if the same is true for polynomials in \(d\) variables, when now, of course \(\mu\) is a determinate measure on \(R^ d\).
The paper settles this question in the negative by proving that there exist rotation invariant measures \(\mu\) on \(R^ d\), \(d>1\), which are determinate, but the polynomials are not dense in \(L^ 2(\mu)\). The paper shows that such measures necessarily have the representation \(\mu=\sum^ \infty_{n=0}\alpha_ n\omega_{r_ n}\), where \(\alpha_ n>0\) and \(\omega_ r\) is the normalized uniform distribution on the sphere \(\| x\|=r\), and \(0\leq r_ 0<r_ 1<\cdots\to\infty\) are the zeros of an entire function of order \(\leq 2\).
Thus for all other rotation invariant determinate measures on \(R^ d\) and in particular, for such a measure \(\mu\) which is absolutely continuous with respect to Lebesgue measure, the polynomials are dense in \(L^ 2(\mu)\).

MSC:

44A60 Moment problems
28B05 Vector-valued set functions, measures and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akhiezer, N. I.,The Classical Moment Problem. Oliver and Boyd, Edinburgh, 1965. · Zbl 0135.33803
[2] Berg, C. &Christensen, J. P. R., Density questions in the classical theory of moments.Ann. Inst. Fourier, 31 (1981), 99–114. · Zbl 0437.42007
[3] Berg, C., Christensen, J. P. R. &Ressel, P.,Harmonic Analysis on Semigroups, Theory of Positive Definite and Related Functions. Graduate Texts in Mathematics, Springer, Berlin-Heidelberg-New York, 1984. · Zbl 0619.43001
[4] Bourbaki, N.,Intégration. Herrmann, Paris, 1959.
[5] Buchwalter, H &Cassier, G., La paramétrisation de Nevanlinna dans le problème des moments de Hamburger.Exposition. Math., 2 (1984), 155–178. · Zbl 0575.44005
[6] Chihara, T. S., Indeterminate symmetric moment problems.J. Math. Anal. Appl., 85 (1982), 331–346. · Zbl 0485.44004 · doi:10.1016/0022-247X(82)90005-1
[7] Fuglede, B., The multidimensional moment problem.Exposition. Math., 1 (1983), 47–65. · Zbl 0514.44006
[8] Havin, V. P. et al. (editors),Linear and complex analysis problem book. 199research problems. Lecture Notes in Mathematics, 1043. Springer, Berlin-Heidelberg-New York, 1984.
[9] Heyde, C. C., Some remarks on the moment problem I.Quart. J. Math. Oxford (2), 14 (1963), 91–96. · Zbl 0112.10005 · doi:10.1093/qmath/14.1.91
[10] Nelson, E., Analytic vectors.Ann. Math. (2), 70 (1957), 572–615. · Zbl 0091.10704 · doi:10.2307/1970331
[11] Nevanlinna, R., Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjessche Momentenproblem.Ann. Acad. Sci. Fenn. Ser. AI, 18.5 (1922), (52 pp.). · JFM 48.1226.02
[12] Nussbaum, A. E., A commutativity theorem for unbounded operators in Hilbert space.Trans. Amer. Math. Soc., 140 (1969), 485–491. · Zbl 0181.40905 · doi:10.1090/S0002-9947-1969-0242010-0
[13] Petersen, L. C., On the relation between the multidimensional moment problem and the one-dimensional moment problem.Math. Scand., 51 (1982), 361–366. · Zbl 0514.44007
[14] Riesz, M., Sur le problème des moments et le théorème de Parseval correspondant.Acta Litt. Ac. Sci. Szeged, 1 (1923), 209–225. · JFM 49.0708.02
[15] Schmüdgen, K., On determinacy notions for the two dimensional moment problem. To appear inArk. Mat. · Zbl 0762.44004
[16] Stein, E. M. &Weiss, G.,Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, 1971. · Zbl 0232.42007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.