PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 3, March 1998, Pages 761–768 S 0002-9939(98)04109-4 VECTOR MOMENT PROBLEMS FOR RAPIDLY DECREASING SMOOTH FUNCTIONS OF SEVERAL VARIABLES RICARDO ESTRADA (Communicated by J. Marshall Ash) Abstract. The existence of rapidly decreasing smooth solutions of moment problems for functions of several variables with values in a Fr´ echet space is obtained. It is shown that the corresponding results for functions with values in a general topological vector space do not hold. 1. Introduction The problem of moments is an important mathematical problem which has at- tracted the attention of mathematicians for over a century. The moments have been shown to be of importance in several areas of the classical analysis [1], [14], while recent developments have shown that they also play a prominent role in ar- eas of current interest such as the asymptotic expansion of generalized functions [4], [5], [7], [8], the theory of orthogonal polynomials [10], [12] and the theory of distributional solutions of differential equations [11], [16]. The aim of the present article is to study the vector moment problem for func- tions of several variables  R n f (x) x k dx = a k , k ∈ N n , (1.1) where {a k } k∈N n is a net of elements of a topological vector space E and where f : R n → E is a rapidly decreasing smooth function. Notice the standard notation x k = x k1 1 ...x kn n if x =(x 1 ,...,x n ) and k =(k 1 ,...,k n ). We show that when E is a Fr´ echet space, then (1.1) has rapidly decreasing smooth solutions for arbitrary nets {a k }. Actually, if V is an open cone in R n , then (1.1) has solutions with support in V . In the very important particular case when E is R or C we obtain that if {μ k } is an arbitrary net of real or complex numbers, then the moment problem  V φ(x) x k dx = μ k , k ∈ N n , has solutions φ ∈S (R n ) with supp φ ⊆ V . Received by the editors May 25, 1995 and, in revised form, September 3, 1996. 1991 Mathematics Subject Classification. Primary 30E05, 46F40. c 1998 American Mathematical Society 761 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use