Polynomial Approximation on Unbounded Subsets and the Moment Problem
In the first part of this work, one proves a Markov moment problem involving L1– norm on a space Lν1 (R+n) for a regular positive special measure ν.. To this end, polynomial approximation on unbounded subsets and Hahn – Banach principle are applied. One uses approximation by sums of tensor products of positive polynomials in each separate variable. This way, one solves the difficulty created by the fact that there are positive polynomials, which are not writable as sums of squares in several dimensions. Consequently, we can solve the multidimensional moment problem in terms of quadratic mappings. We also discuss Markov moment problems in concrete spaces. These last results represent interpolation problems with two constraints. Here the main ingredients of the proofs are constrained extension theorems for linear operators.