Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients
Monic orthogonal polynomials on the unit circle are completely characterized by the Szegő recurrence relation \(P_{n+1}(z) = zP_n(z) - \bar{b_n} P_n^*(z)\), where \(b_n\) \((n=0,1,2,\ldots)\) are known as the reflection coefficients (Schur parameters) and \(P_n^*(z) = z^n \bar{P}_n(1/z)\) are the reversed polynomials. In the present paper the authors investigate the situation when \(\lim_{n \to \infty} b_{nN+j} = a_j\), where \(a_{j+N}=a_j\). First they investigate the polynomials with the periodic reflection coefficients \(a_n\) \((n=0,1,2,\ldots)\) and then they study the asymptotic periodic case as a perturbation of the periodic case. They obtain comparative asymptotics on and off the support of the orthogonality measure (which is essentially the union of several arcs on the unit circle) and give a description of the (asymptotic) location of the zeros. Finally, they also have results regarding the absolutely continuous part of the measure and they show that there are at most finitely many mass points if \(b_n-a_n\) converges geometrically fast to zero.
orthogonal polynomials on the unit circle
scientific article; zbMATH DE number 1006709
scientific article