We apply the theory of holomorphic functions of several complex
variables to prove a multidimensional variant of an exponential
boundedness  criterion \cite{BergChr} for the classical  moment
problem and we extend a theorem of Petersen \cite{Petersen}
concerning the relation between the multidimensional moment problem
and the one-dimensional moment problem for half-lines and compact
subsets of the real line $\mathbb R$. We use these results to solve
the moment problem for the  quantum phase space observables
generated by the number states.
