Indeterminate symmetric moment problems

Abstract

Let {n,(x)} be a sequence of orthogonal polynomials with respect to a real distribution &. According to a classical theorem of Reisz 191, (x,(x)] is complete in L: if and only if w is either the solution of a determined Hamburger moment problem or an extremal solution of an indeterminate moment problem. In the special case where the zeros of the orthogonal polynomials have a finite infimum a and the Hamburger moment problem is indeterminate, there is an extremal solution which is essentially the only solution whose spectrum is a subset of [a, co) [4]. This solution is, in a sense, the “best” solution from the viewpoint of orthogonal polynomials since its spectrum consists of the limit points of the zeros of the polynomials. Askey remarked to the author that in view of Riesz’ theorem, it is surprising that so little attention has been paid to the question of extremal solutions. (Moak [8] has recently studied the extremal solutions associated with the “q-Laguerre,” nee generalized Stieltjes-Wigert, polynomials.) Askey suggested it would be desirable to investigate the existence of “best” extremal solutions in the general case or, as a first step, even in the “symmetric case” (moments of odd order are 0). In this paper, we will consider this special case and the related systems of orthogonal polynomials. In particular, we will show that an indeterminate symmetric moment problem has precisely two “symmetric” extremal solutions. This result, which we found surprizing at first, appears after reflection quite natural (and perhaps obvious) since the two solutions are the symmetrizations of the “best” extremal solutions of the related moment problems on [0, co).