Vector spaces of functions with mostly real zeros (Q1342484)

A theorem of Kreĭn states that a real meromorphic function mapping the upper half plane onto itself must have the form \[ c\frac{z-a_0}{z- b_0} \prod \Pi \left( 1-\frac{z}{a_k} \right) \left( 1- \frac{z}{b_k} \right)^{-1}, \] where \(b_k< a_k< b_{k+1}\), \(a_{-1}< 0< b_1\), \(c>0\) and the product may be finite, one-, or two- tailed. In this paper Kreĭn's theorem is given an algebraic interpretation, and a stronger theorem is proved.