Rakhmanov's theorem for orthogonal matrix polynomials on the unit circle
نویسنده
چکیده
Rakhmanov’s theorem for orthogonal polynomials on the unit circle gives a sufficient condition on the orthogonality measure for orthogonal polynomials on the unit circle, in order that the reflection coefficients (the recurrence coefficients in the Szegő recurrence relation) converge to zero. In this paper we give the analog for orthogonal matrix polynomials on the unit circle. 1. Rakhmanov’s theorem in the scalar case Let φn(z) = κnz n + · · · (n = 0, 1, 2, . . . ), with κn > 0, be orthonormal polynomials on the unit circle with respect to some positive measure μ: ∫ 2π 0 φn(z)φm(z) dμ(θ) = δm,n, z = e . We denote the monic polynomials by Φn(z) = φn(z)/κn. These monic polynomials satisfy a useful recurrence relation (1.1) Φn(z) = zΦn−1(z) + Φn(0)Φ ∗ n−1(z), where Φ∗n(z) = z Φn(1/z) is the reversed polynomial (see, e.g., [12,15]). The coefficients Φn(0), which act as recurrence coefficients in this recurrence relation, are known as reflection coefficients and αn = −Φn+1(0) are called Verblunsky coefficients in [12]. It is well known that all the zeros zk,n of φn lie in the open unit disk, and hence |Φn(0)| = ∏n k=1 |zk,n| < 1. Moreover, (1.2) κn−1 κn = 1− |Φn(0)| , so that the reflection coefficients allow us to compute the monic orthogonal polynomials recursively using (1.1), but also the orthonormal polynomials using in addition 1991 Mathematics Subject Classification. 42C05.
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عنوان ژورنال:
- Journal of Approximation Theory
دوره 146 شماره
صفحات -
تاریخ انتشار 2007