## 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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