An indeterminate rational moment problem and Carathéodory functions

Let { α n } n = 1 ∞ be a sequence of points in the open unit disk in the complex plane and let B 0 = 1 and B n ( z ) = ∏ k = 0 n α k ¯ | α k | α k - z 1 - α k ¯ z , n = 1 , 2 , … , ( α k ¯ / | α k | = - 1 when α k = 0 ). We put L = span { B n : n = 0 , 1 , 2 , … } and we consider the following “moment” problem: a positive-definite Hermitian inner product 〈 · , · 〉 in L , find all positive Borel measures ν on [ - π , π ) such that 〈 f , g 〉 = ∫ - π π f ( e i θ ) g ( e i θ ) ¯ d ν ( θ ) for f , g ∈ L . We assume that this moment problem is indeterminate. Under some additional condition on the α n we will describe a one-to-one correspondence between the collection of all solutions to this moment problem and the collection of all Carathéodory functions augmented by the constant ∞ .