Abstract

this paper to ll this gap. To state our theorem we use the notation lim n=N!t X n;N = X 4 Kuijlaars and Van Assche to denote the property that in the doubly indexed sequence X n;N we have lim j!1 X n j ;N j = X whenever n j and N j are two sequences of natural numbers such that N j ! 1 and n j =N j ! t as j !1. For example, the convergence in Proposition 1.3 may be expressed by lim n=N!t (p n;N ) = w;t : We will use this notation throughout the rest of the paper. Our main result is the following. Theorem 1.4 Let for each N 2 N, two sequences fa n;N g 1 n=1 , a n;N > 0 and fb n;N g 1 n=0 of recurrence coecients be given, together with orthogonal polynomials p n;N generated by the recurrence xp n;N (x) = a n+1;N p n+1;N (x) + b n;N p n;N (x) + a n;N p n 1;N (x); n 0; (1.6) and the initial conditions p 0;N 1 and p 1;N 0. Suppose that there exist two continuous functions a : (0; 1) ! [0; 1), b : (0; 1) ! R, such that lim n=N!t a n;N = a(t); lim n=N!t b n;N = b(t) (1.7) whenever t > 0. Dene the functions (t) := b(t) 2a(t); (t) := b(t) + 2a(t); t > 0: (1.8) Then we have for every t > 0, lim n=N!t (p n;N ) = 1 t Z t 0 ! [(s);(s)] ds: (1.9) Here ! [;] is the measure given by (1.4) if < . If = , then ! [;] is the Dirac point mass at . Remark 1.5 The measure on the right-hand side of (1.9) is the average of the equilibrium measures of the varying intervals [(s); (s)] for 0 < s < t. Its support is given by " inf 0<s<t (s); sup 0<s<t (s) # : (1.10) In particular, the support is always an interval. The support is unbounded if or are unbounded near 0. J. Approx. Theory 99 (1999), 167-197. 5 Remark 1.6 Theorem 1.4 has an obvious extension to polynomials that are orthogonal with respect to a discrete measure supp...

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