Extensions of Szegö's theory of rational functions orthogonal on the unit circle
Let \(S= \{z_{nk}\}\), \(k= 1,\dots, n\), \(n= 1, 2,\dots,\) be a triangular array of points in the open unit disk \(D\) of the complex plane. Define \(B_{n0}= 1\) and \(B_{nk}\) the Blaschke product with zeros \(z_{nk}\), \(k= 1,\dots, n\) and \({\mathcal L}_n= \text{span}\{B_{nk}, k= 0,\dots, n\}\). If \(\mu\) is a positive Borel measure on \([0, 2\pi)\), let \(\phi_{nk}\), \(k= 0,\dots, n\), be obtained by orthogonalization of the basis \(B_{nk}\) in \(L^2(d\mu)\), the Hilbert space with inner product \((f, g)= \int^{2\pi}_0 f(e^{i\theta}) \overline{g(e^{i\theta})} d\mu(\theta)\). The rational functions \(\phi_{nk}\) generalize the Szegö polynomials \(\phi_k\) in the sense that if all \(z_{nk}= 0\), then \(\phi_{nk}= \phi_k\). This paper studies the convergence of \(\phi_{nk}\) as \(n\to \infty\) while it is assumed that \(\mu'> 0\) a.e. and \(S\) is compactly contained in \(D\). If \(\phi_n\) is a Szegö polynomial and \(\phi^*_n(z)= z^n \overline{\phi_n(1/\overline z)}\) its reciprocal, then generalizations of the following well-known polynomial results are obtained: \(\phi^*_n/\phi_n\to 0\) locally uniformly in \(|z|> 1\) and \(\phi^*_{n- 1}/\phi^*_n\to 1\) l.u. in \(D\). Several \(L^2(d\theta)\) and \(L^1(d\theta)\) convergence results are also included. The generalization of the polynomial result \(|\mu_n- 1|_{L^2(d\theta)}\to 0\) with \(\mu_n= |\phi_n|^2 \mu'\) is but one example. The proofs rely on earlier work by the author, on results by G. López for orthogonal polynomials w.r.t. varying measures and related results obtained by A. Bultheel et al.
scientific article; zbMATH DE number 903481