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 [M. Putinar, F.-H. Vasilescu, Solving moment problems by dimensional extension, Ann. of Math. (2) 149 (3) (1999) 1087-1107] for the corresponding multidimensional moment problem.
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Assume now that Λ(pk|r|2) 0 for all r ∈ R and 1 ⩽ k ⩽ m. This condition is equivalent to the operators P1,...,Pm being positive. We recall that these operators are essentially selfadjoint. But for all such k, Pk ⊆ pk(A) by Claim 1, and pk(A) is selfadjoint; thus, Pk = pk(A) is a positive selfadjoint operator. Equivalently, its spectral measure is supported by R+. But the spectral measure of pk(A) is Fk(δ) = E(pk−1(δ)). Hence, E itself is supported⋂ by pk−1(R+). Since that is true for all 1 ⩽ k ⩽ m, the support of E is therefore a subset of k=1mpk−1(R+).
- Laplace transform
- Multidimensional fractional moment problem
- Widder Theorem
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