Favard theorem for reproducing kernels

Consider for n = 0, 1, … the nested spaces Ln of rational functions of degree n at most with given poles 1α̇i, ¦αi¦ < 1, i = 1, …, n. Let L = ∪0∞Ln. Given a finite positive measure μ on the unit circle, we associate with it an inner product on L by 〈ƒ,g〉 = ∫ ƒḡ dμ . Suppose kn(z, w) is the reproducing kernel for Ln, i.e., 〈ƒ(z),kn(z,w)〉 = ƒ(w), for all ƒ ∈ L n, ¦w¦ < 1, then it is known that they satisfy a coupled recurrence relation. s paper we shall prove a Favard type theorem which says that if you have a sequence of kernel functions kn(z, w) which are generated by such a recurrence, then there will be a measure μ supported on the unit circle so that kn is the reproducing kernel for Ln. The measure is unique under certain extra conditions on the points αi.