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Zbl 0837.44007
Aizenberg, Lev; Zalcman, Lawrence
Instability phenomena for the moment problem. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22, No.1, 95-107 (1995). ISSN 0391-173X

Let $\mu$ be a finite complex Borel measure on a compact set $K \subset \bbfC^n$, $n \ge 1$, and let $a_\alpha = \int {d \mu \over \zeta^{\alpha + I}}$ be its moments, where $\alpha = (\alpha_1, \dots, \alpha_n)$, $I = (1, \dots, 1)$, $\zeta^\alpha = \zeta_1^{ \alpha_1}\cdot \dots \cdot \zeta_n^{\alpha_n}$. Under some conditions on $K$, the inequality $$ \varlimsup_{|\alpha |\to \infty} {\root |\alpha |\of {|a_\alpha |d_\alpha (K)}} < 1,$$ where $d_\alpha (K) = \max_K |z^\alpha |$, is shown to imply $a_\alpha = 0 $, $\forall \alpha \in (\bbfZ_+)^n$. In some cases it is then possible to conclude that $\mu = 0$. \par A similar result is obtained for harmonic moments of a measure on a compact set $K \subset \bbfR^n$, $n \ge 2$.
[A.Yu.Rashkovsky (Khar'kov)]

MSC 2000:: *44A60 Moment problems
31B15 Potentials, etc. (higher-dimensional)

Keywords: moment problem; uniqueness theorem; complex Borel measure; harmonic moments

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