On the problem of modified moments

Abstract:

The problem of modified moments is studied. Let $\left ( {{P_n}\left ( x \right )} \right )_n^\infty = 0$ be an orthogonal polynomial sequence. Given a sequence $\left ( {{d_n}} \right )_n^\infty = 0$ of real numbers, does there exist a bounded nondecreasing function with infinitely many points of increase such that for every $n \in {{\mathbf {N}}_0}$, ${d_n} = \int _{ - \infty }^\infty {{P_n}} (x)d\mu (x)$? Is there any information about the support of $\mu$? A necessary and sufficient condition for the existence of such a function $\mu$ is given in terms of the positivity of certain determinants. For certain $\left ( {{P_n}\left ( x \right )} \right )_{n = 0}^\infty$ a description of the support of $\mu$ is established.