Favard theorem for reproducing kernels
Consider for \(n = 0, 1, \dots\) the nested spaces \({\mathcal L}_n\) of rational functions of degree \(n\) at most with given poles \(1/ \overline \alpha_i\), \(|\alpha_i |< 1\), \(i = 1, \dots, n\). Let \({\mathcal L} = \cup^\infty_0 {\mathcal L}_n\). Given a finite positive measure \(\mu\) on the unit circle, we associate with it an inner product on \({\mathcal L}\) by \(\langle f,g \rangle = \int f \overline gd \mu\). Suppose \(k_n (z,w)\) is the reproducing kernel for \({\mathcal L}_n\) i.e., \(\langle f(z), k_n (z,w) \rangle = f(w)\), for all \(f \in {\mathcal L}_n\), \(|w |< 1\), 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 \({\mathcal L}_n\). The measure is unique under certain extra conditions on the points \(\alpha_i\).
scientific article; zbMATH DE number 770335
scientific article