FAVARD THEOREM FOR REPRODUCING KERNELS

Consider for n = 0, 1, ... the nested spaces L(n), of rational functions of degree n at most with given poles 1/<(alpha)over bar>(i), \alpha(i)\ < 1, i = 1,..., n. Let L = boolean OR(0)(infinity) L(n). Given a finite positive measure mu on the unit circle, we associate with it an inner product on L by (f,g) = integral f (g) over bar d mu. Suppose k(n)(z, w) is the reproducing kernel for L(n), i.e., [f(z), k(n)(z, w)] = f(w), for all f is an element of 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 mu 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 alpha(i).