Rotation invariant moment problems

An important theorem of Marcel Riesz, cf. [14], states that the polynomials are dense in L2(/x), when/x is a determinate measure on the real line. In the indeterminate case Riesz also characterized the measures/x for which the polynomials are dense in L2(#). They are the so-called Nevanlinna extremal measures, introduced in Nevanlinna [11]. It does not seem to be known whether the polynomials are dense in L2(g), when/x is a determinate measure on R d, d> 1, cf. the expository paper by Fuglede [7], as well as the research problems book [8, p. 529], where Devinatz poses the problem as question 1 and ascribes it to the physicist John Challifour (1978). In this paper we shall settle the question in the negative. There exist rotation invariant measures # on R d, d> 1, which are determinate but for which the polynomials are not dense in L2(p). Such measures/x are necessarily of the following very special form