The Sobolev moment problem and Jordan dilations

Moment problems and orthogonal polynomials, both meant in a single real variable, belong to the oldest problems in Classical Analysis. They have been developing for over a century in two parallel, mostly independent streams. During the last 20 years a rapid advancement of polynomials orthogonal in Sobolev space has been noticed, see [5] for an updated survey; their moment counterpart seems not to be paid enough attention as it deserves. In this paper we intend to resume the theme of [3] and also [4], and open the door for further, deeper study of the problem.