Moments from their very truncations

It is known that positive definiteness is not enough for the multidimensional moment problem to be solved. We would like throw in to the garden of existing in this matter so far results one more, a result which takes into considerations the utmost possible truncations. As we have already pointed out positive definiteness is not sufficient for a multisequence to be a moment one, neither in the case of real moment problem in more than one variable, nor for a complex one or any complex dimension; for the previous one we recommend the cult paper of Fuglede [5], for the second mentioned [8] can be regarded as a source of information. Replacing it by solvability of a kind of truncations we gain necessary and sufficient conditions for the moment problem to be settled. It is worthy to say that truncations in the multivariable development have been considered from diverse points of view; let us have [3], [4], [7], [16] or [17] as a choice of references. 1. Let M(X) stand for the space of all regular complex Borel measures on a locally compact space X and letMa(X) be the collection of all positive measures in M(X) such that μ(X) = a. Consider M(X) with the σ(M(X), Cb(X)) topology , where Cb(X) is the Banach space of continuous and bounded functions on X with the ‘sup’ norm; the topology is determined by the duality (μ, f) 7→ ∫ X f dμ, μ ∈ M(X), f ∈ Mb(X). One of the pleasant features of the σ(M(X), Cb(X)) topology is that Ma(X) is stable under the closure, another is that it coincides on Ma(X) with the ∗–weak topology. 1991 Mathematics Subject Classification. Primary 44A60; Secondary 43A35, 43A05, 47B32, 47B15, 47B20.