Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights. (English) Zbl 1050.44003

A positive Borel measure \(\mu\) on \({\mathbb R}^n\) such that \(\int_{{\mathbb R}^n} \| x\| ^d \,d\mu(x)<\infty\) for all \(d\geq 0\) is said to be determinate (in the Hamburger sense) if \(\mu\) is uniquely determined by its integrals against the polynomials on \({\mathbb R}^n\). Here it is proved in a direct way that a measure is determinate if the multi-dimensional analogue of Carleman’s condition is satisfied. In that case, the polynomials are dense in the corresponding \(L_p\)-spaces for \(1\leq p<\infty\), and so is \(\text{Span}_{\mathbb C}\{e_ {i\lambda}:\lambda\in S\}\) for any \(S\subset {\mathbb R}^n\) with the property that its closure has nonempty interior.
Sufficient conditions for Hamburger determinacy imply sufficient conditions for determinacy in the Stieltjes sense, which amounts to ask whether a measure on \({\mathbb R}^n\) is determined by its integrals against the polynomials under the assumption that its support is contained in a given positive convex cone with the origin as vertex.
From these results, integral criteria for determinacy both in the Hamburger and in the Stieltjes sense which involve quasi-analytic weights are derived.


44A60 Moment problems
26E10 \(C^\infty\)-functions, quasi-analytic functions
41A10 Approximation by polynomials
41A63 Multidimensional problems
42A10 Trigonometric approximation
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: DOI arXiv Euclid


[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. · Zbl 0919.60005
[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. · Zbl 0437.42007
[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. · Zbl 0531.28008
[5] CHIHARA, T. S. (1968). On indeterminate Hamburger moment problems. Pacific J. Math. 27 475-484. · Zbl 0167.41901
[6] DE JEU, M. F. E. (2001). Subspaces with equal closure. Constr. Approx.
[7] DUNFORD, N. and SCHWARTZ, J. T. (1957). Linear Operators. Part I. Wiley, New York. · Zbl 0635.47001
[8] FUGLEDE, B. (1983). The multidimensional moment problem. Expo. Math. 1 47-65. · Zbl 0514.44006
[9] HOFFMAN-JORGENSEN, J. (2002). The moment problem.
[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. · Zbl 0931.30001
[12] LIN, G. D. (1997). On the moment problems. Statist. Probab. Lett. 35 85-90. · Zbl 0904.62021
[13] NUSSBAUM, A. E. (1966). Quasi-analytic vectors. Ark. Mat. 6 179-191. · Zbl 0182.46102
[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. · Zbl 0997.60001
[15] PETERSEN, L. C. (1982). On the relation between the multidimensional moment problem and the one-dimensional moment problem. Math. Scand. 51 361-366. · Zbl 0514.44007
[16] SHOHAT, J. and TAMARKIN, J. D. (1943). The Problem of Moments. Amer. Math. Soc., Providence, RI. · Zbl 0063.06973
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.