Favard theorem for reproducing kernels

Abstract

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