This review paper deals with recalling some applications of polynomial approximation results to Markov moment problem as well as to invariance of the unit ball of L^1 -type spaces, with respect to some bounded linear operators. Polynomial approximation on special unbounded subsets is mainly discussed. One solves partially the difficulty created by the fact that in several real dimensions positive polynomials are not sums of squares. Most of our characterizations are expressed in terms of signatures of products of quadratic mappings. Main such results were published by the authors in the period 2007-2015. Some applications of these theorems, which have been published more recently, are also recalled. Earlier results on extension of linear operators with one or two constraints are applied as well.
