Asymptotics of derivatives of orthogonal polynomials on the unit circle

We show that ratio asymptotics of orthogonal polynomials on the circle imply ratio asymptotics for all their derivatives. Moreover, by reworking ideas of P. Nevai, we show that uniform asymptotics for orthogonal polynomials on an arc of the unit circle imply asymptotics for all their derivatives. Let be a …nite positive Borel measure on the unit circle (or [0; 2 ]). Let f'ng denote the orthonormal polynomials for , so that 1 2 Z 2 0 'n e 'm e d ( ) = mn: Asymptotics for derivatives of orthogonal polynomials have been established under various hypotheses [1], [2], [5], [6], [8]. As far as the author is aware, the result that applies to the most general weights is due to P. Nevai [6]. Assuming Szeg1⁄2o’s condition Z 2 0 log 0 ( ) d > 1; and for some 2 (0; 2 ) and > 0, is absolutely continuous in ( ; + ), with Z + 0 ( ) 0 (t) t !2 dt <1; (1.1) Nevai [6] proved that for each m 1, lim n!1 z' n (z) = (n 'n (z)) = 1; where z = e . If the condition (1.1) holds uniformly for in an interval I, then the asymptotic holds uniformly for in I. Nevai’s proof involved similar ? Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353. Email address: [email protected] (D. S. Lubinsky). Preprint submitted to Elsevier Science 9 October 2006 techniques to those for proving the asymptotics of f'ng themselves, which are in turn equivalent to asymptotics for the reversed polynomials ' n (z) = z 'n (1=z): (1.2) By reworking Nevai’s ideas, we show that uniform asymptotics of f' ng on an arc directly imply asymptotics for derivatives of 'n. Theorem A. Let J be a subinterval of [0; 2 ], and assume that lim n!1 ' n e = g ( ) ; uniformly for 2 J , where g ( ) 6= 0 for 2 J . Let m 1 and I J be a closed interval. Then uniformly for z = e , 2 I, lim n!1 z' n (z) = (n 'n (z)) = 1: Proof. Note …rst that because of the uniform convergence, g is continuous, and hence jgj is bounded away from 0 in any compact subinterval of J . Let I 0 be a compact subinterval of J such that (I 0) I. Because of uniform convergence, for some constant C independent of n, k' nkL1(I0) := sup 2I ' n e C; n 1: We now follow an idea of Nevai [6]. Let [ p n] denote the integer part of p n. By uniform convergence, n = ' n ' [n] L1(I0) ! 0; n!1. We shall apply Markov-Bernstein inequalities for trigonometric polynomials on arcs I 0 and their proper subarcs I. Let D = d d . If R is a trigonometric polynomial of degree n, and ` 1 [3, pp. 242–243] DR L1(I) C1n kRkL1(I0) ; where C1 depends only `; I; I 0. This and the above bounds give D' n L1(I) D ' n ' [n] L1(I) + D' [n] L1(I) C1n n + C1 p n ` C = o n : (1.3) Next, we apply Leibniz’s formula to the identity 'n e = e ' n (e i ); (1.4)