Moment Problems on Bounded and Unbounded Domains

Using approximation results, we characterize the existence of the solution for a two-dimensional moment problem in the first quadrant, in terms of quadratic forms, similar to the one-dimensional case. For the bounded domain case, one considers a space of complex analytic functions in a disk and a space of continuous functions on a compact interval. The latter result seems to give sufficient (and necessary) conditions for the existence of a multiplicative solution. 1. Introduction Applying the extension Hahn-Banach type results in existence questions concerning the moment problem is a well-known technique [1–11]. One of the most useful results is lemma of the majorizing subspace (see [12, Section 5.1.2] for the proof of the lattice-version of this lemma; see also [13]). It says that if is a linear positive operator on a subspace of the ordered vector space , the target space being an order complete vector lattice , and for each there is , then has a linear positive extension . Another geometric remark is that in the real case, the sublinear functional from the Hahn-Banach theorem can be replaced by a convex one. The theorem remains valid when the convex dominating functional is defined on a convex subset with some qualities with respect to the subspace (for instance, ri ), where ri is the relative interior of ). Here we recall an answer published without proof in 1978 [14], without losing convexity, but strongly generalizing the classical result. The first detailed proof was published in 1983 [15]. The proof of a similar result, in terms of the moment problem, was published in [10]. Here we recall the general statement from [14]. One of the reasons is that many other results are consequences of this theorem, including Bauer’s theorem [13], Namiokas’s theorem, and abstract moment problem-results published firstly in [9]. Part of these generalizations of the Hahn-Banach principle are applied in the present work too. Throughout this first part, will be a real vector space, an order-complete vector lattice, convex subsets, a concave operator, a convex operator, a vector subspace, and a linear operator. Theorem 1. Assume that The following assertions are equivalent:(a) there is a linear extension of the operator such that (b) there are convex and concave operators such that for all ？？one has The minus-sign appears to construct a convex operator in the left-hand side member and a concave operator in the right side. The idea of sandwich theorem on arbitrary convex subsets is clear. Most of the applications hold for linear positive operators on linear ordered