Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients. II: Weak asymptotics
Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients \(a_n = P_{n+1}(0)\) which appear as recurrence coefficients in the recurrence relation for the monic polynomials \(P_n(z)\). In this paper the authors assume that the reflection coefficients are asymptotically periodic with period \(N\), i.e., \(\lim_{k \to \infty} a_{kN+j} = a_j^0\) for \(j=0,1,\ldots,N-1\). First, the orthogonal polynomials with periodic reflection coefficients are analyzed in Section 2, where known results of Geronimus are recalled. The support of the orthogonality measure consists of at most \(N\) arcs on the unit circle and at most a finite number of mass points outside these arcs. Then, in Section 3, the orthogonal polynomials with asymptotically periodic reflection coefficients are investigated, using the periodic case as a comparison system. The results in this paper deal with weak asymptotics, i.e., the asymptotics of integrals of the form \(\int g(\theta) |\Phi_{kN+j}(e^{i\theta})|^2 d\sigma\) when \(k \to \infty\) and \(\Phi_n\) the orthonormal polynomials (Theorem 3). The authors also deal with weak asymptotics of the Christoffel function (Theorem 4). [For Part I see J. Approximation Theory 88, No. 3, 316-353 (1997; Zbl 0889.42021)].
orthogonal polynomials on the unit circle
scientific article; zbMATH DE number 1495765
scientific article