Generalized Widder Theorem via fractional moments

We provide a necessary and sufficient condition for the representability of a function as the classical multidimensional Laplace transform, when the support of the representing measure is contained in some generalized semi-algebraic set. This is done by employing a method of Putinar and Vasilescu [Putinar, M. and Vasilescu, F.-H., Solving moment problems by dimensional extension, Ann. of Math. (2) 149 (1999), no. 3, 1087-1107] for the corresponding multidimensional moment problem.