Necessary and sufficient conditions for the existence of the solutions of a class of scalar and mainly for operator-valued moment problems are reviewed. This was the first motivation for proving our constrained extension results for linear operators. Polynomial approximations on bounded and on unbounded closed subsets are very useful in proving the uniqueness of the solution. We also reviewed earlier results on the extension of positive linear functional and operators. Such results are applied to ensure the extension of our linear solution from the subspace of polynomials to a larger function space. In most of the cases from below, this is made using polynomial approximation in one and several variables. Besides positivity, our solution is bounded from above by a dominating linear, sublinear or only convex continuous operator, on the entire domain space or only on its positive cone. This allows estimating the norm of the linear solution.
