On the problem of modified moments

The problem of modified moments is studied. Let (P_n(x))^∞_n=0 be an orthogonal polynomial sequence. Given a sequence (d_n)^∞_n=0 of real numbers, does there exist a bounded nondecreasing function with infinitely many points of increase such that for every n ∈ N₀, d_n = ∫^∞_-∞ P_n(x) dμ(x)? Is there any information about the support of μ? A necessary and sufficient condition for the existence of such a function v is given in terms of the positivity of certain determinants. For certain (P_n(x))^∞_n=0 a description of the support of v is established.