Abstract
We consider the problem of finding a (non-negative) measure μ on B(Cn) such that ∫Cnzkdμ(z)=sk, ∀k∈K. Here, K is an arbitrary finite subset of Z+n, which contains (0,…,0), and sk are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. Firstly, one may consider this problem as an extension of the truncated multidimensional moment problem on Rn, where the support of the measure μ is allowed to lie in Cn. Secondly, the moment problem is a particular case of the truncated moment problem in Cn, with special truncations. We give simple conditions for the solvability of the above moment problem. As a corollary, we have an integral representation with a non-negative measure for linear functionals on some linear subspaces of polynomials.
Highlights
We denote by R, C, N, Z, Z+ the sets of real numbers, complex numbers, positive integers, integers and non-negative integers, respectively
There are two possible interpretations of this problem. One may consider this problem as an extension of the truncated multidimensional moment problem on Rn, where the support of the measure μ is allowed to lie in Cn
In [11], we presented the operator approach to the truncated multidimensional moment problem in
Summary
These truncations do not include conjugate terms It is well-known that the multidimensional moment problems are much more complicated than their one-dimensional prototypes [1,4,5,6,7,8]. In [11], we presented the operator approach to the truncated multidimensional moment problem in. A detailed exposition of the theory of (full and truncated) multidimensional moment problems is given in a recent Schmüdgen’s book [8]. In the case of the moment problem (1), we shall need a modification of the operator approach, since we have no positive definite kernels here. This problem can be passed and we shall come to some commuting bounded operators.
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