No abstract

The paper is a continuation of work initiated by the first two authors in [K-M]. Section 1 is introductory. In Section 2 we prove a basic lemma, Lemma 2.1, and use it to give new proofs of key technical results of Scheiderer in [S1] [S2] in the compact case; see Corollaries 2.3, 2.4 and 2.5. Lemma 2.1 is also used in Section 3 where we continue the examination of the case n = 1 initiated in [K-M], concentrating on the compact case. In Section 4 we prove certain uniform degree bounds for representations in the case n = 1, which we then use in Section 5 to prove that (‡) holds for basic closed semi-algebraic subsets of cylinders with compact cross-section, provided the generators satisfy certain conditions; see Theorem 5.3 and Corollary 5.5. Theorem 5.3 provides a partial answer to a question raised by Schmüdgen in [Sc2]. We also show that, for basic closed semi-algebraic subsets of cylinders with compact cross-section, the sufficient conditions for (SMP) given in [Sc2] are also necessary; see Corollary 5.2(b). In Section 6 we prove a module variant of the result in [Sc2], in the same spirit as Putinar's variant [Pu] of the result in [Sc1] in the compact case; see Theorem 6.1. We apply this to basic closed semi-algebraic subsets of cylinders with compact cross-section; see Corollary 6.4. In Section 7 we apply the results from Section 5 to solve two of the open problems listed in [K-M]; see Corollary 7.1 and Corollary 7.4. In Section 8 we consider a number of examples in the plane. In Section 9 we list some open problems.

The multivariate moment problem is investigated in the general context of the polynomial algebra R[

Abstract. Let K be the basic closed semi-algebraic set in R n defined by some finite set of polynomials S and T , the preordering generated by S. For K compact, f a polynomial in n variables nonnegative on K and real > 0, we have that f + ∈ T . In particular, the K-Moment Problem has a positive solution. In the present paper, we study the problem when K is not compact. For n = 1, we show that the K-Moment Problem has a positive solution if and only if S is the natural description of K (see Section 1). For n ≥ 2, we show that the K-Moment Problem fails if K contains a cone of dimension 2. On the other hand, we show that if K is a cylinder with compact base, then the following property holds:This property is strictly weaker than the one given in Schmüdgen (

It is explained how a locally convex (lc) topology τ on a real vector space V extends to a locally multiplicatively convex (lmc) topology τ on the symmetric algebra S(V ). This allows the application of the results on lmc topological algebras obtained by Ghasemi, Kuhlmann and Marshall to obtain representations of τ -continuous linear function-as integrals with respect to uniquely determined Radon measures µ supported by special sorts of closed balls in the dual space of V . The result is simultaneously more general and less general than the corresponding result of Berezansky, Kondratiev andŠifrin. It is more general because V can be any lc topological space (not just a separable nuclear space), the result holds for arbitrary 2d-powers (not just squares), and no assumptions of quasi-analyticity are required. It is less general because it is necessary to assume that L : S(V ) → R is τ -continuous (not just continuous on each homogeneous part of S(V )).Murray Marshall passed away in May 2015. He worked on this manuscript together with us until the very last days of his life. We lost a collaborator of many years and a wonderful friend. We sorely miss him. (M. Ghasemi, M. Infusino, S. Kuhlmann) 2010 Mathematics Subject Classification. Primary 44A60 Secondary 14P99.