Perturbations in Toeplitz matrices. II. Asymptotic properties

AbstractGiven dμ a finite positive Borel measure on the interval [0, 2π]with an infinite set as its support, (φn(z))n = 0∞ the monic orthogonal polynomials, and (ϑn(z))n = 0∞ the orthonormal polynomials on the unit circle associated with this measure, we determine conditions in the measure and in the parameter t in order that the sequences (φn(t))n = 0∞ and (ϑn(t))n = 0∞ do not belong to l2. This allows us to study the asymptotic behaviour of the ratio (knαn) for the leading coefficients of ϑn(z) and Pn(z) with (Pn(z))n = 0∞ the system of orthonormal polynomials with respect to the inner product 〈P(z),Q(z)〉=12π∫02π P(z)Q(z) dμ(0)+P(t)Q(t), z=ei0, with tϵ