An indeterminate rational moment problem and Carathéodory functions

Let {alpha(n)}(n=1)(infinity) be a sequence of points in the open unit disk in the complex plane and let
B-0 = 1 and B-n(Z) = Pi(n)(k=0) (alpha(k)) over bar/vertical bar alpha(k)vertical bar alpha(k) -
z/1-alpha(k)z, n = 1,2, ...,

((alpha(k)) over bar/vertical bar alpha(k)vertical bar = - 1 when alpha(k) = 0). We put L = span{B-n : n =
0,1,2, ...} and we consider the following "moment" problem: Given a positive-definite Hermitian inner product
<.,.> in L, find all positive Borel measure v on [-pi, pi) such that

< f,g > = integral(pi)(-pi) f(e(i0))<(g(e(i0)))over bar>dv(0) for f,g epsilon L.

We assume that this moment problem is indeterminate. Under some additional condition on the alpha(n) we will describe a one-to-one correspondence between the collection of all solutions to this moment problem and the collection of all Caratheodory functions augmented by the constant infinity.