We consider the problem of finding a (non-negative) measure $\mu$ on $\mathfrak{ B}(\mathbb{ C}^n)$ such that $\int_{ \mathbb{ C}^n} \mathbf{ z}^{ \mathbf{ k}} d\mu(\mathbf{ z}) = s_{ \mathbf{ k}}$, $\forall \mathbf{ k}\in\mathcal{ K}$. Here $\mathcal{ K}$ is an arbitrary finite subset of $\mathbb{ Z}^n_+$, which contains $(0,...,0)$, and $s_{ \mathbf{ k}}$ are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. At first, one may consider this problem as an extension of the truncated multidimensional moment problem on $\mathbb{ R}^n$, where the support of the measure $\mu$ is allowed to lie in $\mathbb{ C}^n$. Secondly, the moment problem is a particular case of the truncated moment problem in $\mathbb{ C}^n$, 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.
Categories: math.FA