The Annals of Probability

Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights

Marcel de Jeu

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We prove in a direct fashion that a multidimensional probability measure $\mu$ is determinate if the higher-dimensional analogue of Carleman's condition is satisfied. In that case, the polynomials, as well as certain proper subspaces of the trigonometric functions, are dense in all associated $L_p$-spaces for $1\leq p<\infty$. In particular these three statements hold if the reciprocal of a quasi-analytic weight has finite integral under $\mu$. We give practical examples of such weights, based on their classification.

As in the one-dimensional case, the results on determinacy of measures supported on $\Rn$ lead to sufficient conditions for determinacy of measures supported in a positive convex cone, that is, the higher-dimensional analogue of determinacy in the sense of Stieltjes.

Article information

Ann. Probab., Volume 31, Number 3 (2003), 1205-1227.

First available in Project Euclid: 12 June 2003

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Zentralblatt MATH identifier

Primary: 44A60: Moment problems
Secondary: 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section) 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10} 42A10: Trigonometric approximation 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26E10: $C^\infty$-functions, quasi-analytic functions [See also 58C25]

Determinate multidimensional measures Carleman criterion $L_p$-spaces multidimensional approximation polynomials trigonometric functions multidimensional quasi-analytic classes quasi-analytic weights.


de Jeu, Marcel. Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights. Ann. Probab. 31 (2003), no. 3, 1205--1227. doi:10.1214/aop/1055425776.

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  • [1] BERG, C. (1995). Recent results about moment problems. In Probability Measures on Groups and Related Structures XI (H. Hey er, ed.) 1-13. World Scientific, River Edge, NJ.
  • [2] BERG, C. (1996). Moment problems and poly nomial approximation. 100 ans après Th. J. Stieltjes. Ann. Fac. Sci. Toulouse Math. 6 (special issue) 9-32.
  • [3] BERG, C. and CHRISTENSEN, J. P. R. (1981). Density questions in the classical theory of moments. Ann. Inst. Fourier (Grenoble) 31 99-114.
  • [4] BERG, C. and CHRISTENSEN, J. P. R. (1983). Exposants critiques dans le problème des moments. C. R. Acad. Sci. Paris Sér. I Math. 296 661-663.
  • [5] CHIHARA, T. S. (1968). On indeterminate Hamburger moment problems. Pacific J. Math. 27 475-484.
  • [6] DE JEU, M. F. E. (2001). Subspaces with equal closure. Constr. Approx. To appear.
  • [7] DUNFORD, N. and SCHWARTZ, J. T. (1957). Linear Operators. Part I. Wiley, New York.
  • [8] FUGLEDE, B. (1983). The multidimensional moment problem. Expo. Math. 1 47-65.
  • [9] HOFFMAN-JORGENSEN, J. (2002). The moment problem. Preprint.
  • [10] HRy PTUN, V. G. (1976). An addition to a theorem of S. Mandelbrojt. Ukraïn. Mat. Zh. 28 841-844. (English translation: Ukrainian Math. J. 28 655-658.)
  • [11] KOOSIS, P. (1988). The Logarithmic Integral. I. Cambridge Univ. Press.
  • [12] LIN, G. D. (1997). On the moment problems. Statist. Probab. Lett. 35 85-90.
  • [13] NUSSBAUM, A. E. (1966). Quasi-analytic vectors. Ark. Mat. 6 179-191.
  • [14] PAKES, A. G., HUNG, W. L. and WU, J. W. (2001). Criteria for the unique determination of probability distributions by moments. Aust. N. Z. J. Stat. 43 101-111.
  • [15] PETERSEN, L. C. (1982). On the relation between the multidimensional moment problem and the one-dimensional moment problem. Math. Scand. 51 361-366.
  • [16] SHOHAT, J. and TAMARKIN, J. D. (1943). The Problem of Moments. Amer. Math. Soc., Providence, RI.