Polynomial Approximation on Unbounded Subsets and the Markov Moment Problem

We start this review paper by recalling some known and relatively recent results in polynomial approximation on unbounded subsets. These results allow approximation of nonnegative continuous functions with compact support contained in the first quadrant by sums of tensor products of positive polynomials in each separate variable, on the positive semiaxes. Consequently, we characterize the existence of solution of a two dimensional Markov moment problem in terms of products of quadratic forms. Secondly, one proves some applications of abstract results on the extension of linear operators with two constraints to the Markov moment problem. Two applications related to this last part are considered.