Lens-shaped regions for strong Stieltjes moment problems (Q1582293)

The author studies the strong Stieltjes moment problem (SSMP): let a double infinite sequence \(\{c_n\}_{-\infty}^{\infty}\) of real numbers be given, find all positive measures \(\mu\) on the non-negative real axis with \[ c_n=\int_0^{\infty}t^n d\mu(t),\quad n=0,\pm 1,\pm 2,\ldots, \] and specifically the case when the problem is indeterminate (more than one solution). The main result is concerned with the value set of the Stieltjes transform of the solutions \[ F(z,\mu)=\int_0^{\infty} {d\mu(t)\over t-z}. \] It is proved that for an arbitrary \(z\) in the complex plane, the values \(F(z,\mu)\) where \(\mu\) ranges over all solutions of the SSMP cover exactly a lens-shaped region. This lens-shaped region is the intersection of two `limiting' disks (following from associated strong Hamburger moment problems -- the measure lives on the whole real axis) generated by sequences of Möbius transforms; these two disks are either really closed disks with positive radius or reduce to a point. This also gives a condition on determinacy of the SSMP: at least one of the disks has to reduce to one point. The paper is well written, though quite technical, and uses the connection with orthogonal Laurent polynomials; an excellent introduction to these polynomials is the book [``Orthogonal rational functions'', Cambridge University Press (1999; Zbl 0923.42017)] by \textit{A. Bultheel}, \textit{P. González-Vera}, \textit{E. Hendriksen}, and \textit{O. Njåstad}.