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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.

MSC:

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.)
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