IMA Journal of Numerical Analysis Advance Access originally published online on January 29, 2008
IMA Journal of Numerical Analysis 2009 29(1):1-23; doi:10.1093/imanum/drm048
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Recurrence and asymptotics for orthonormal rational functions on an interval
Department of Computer Science, Katholieke Universiteit Leuven, Heverlee, Belgium
Email: Karl.Deckers{at}cs.kuleuven.be
Received on 10 April 2007. Revised on 28 November 2007.
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Abstract |
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Let µ be a positive bounded Borel measure on a subset I of the real line and 
= {
1, ..., n} a sequence of arbitrary ‘complex’ poles outside I. Suppose {
1, ..., n} is the sequence of rational functions with poles in orthonormal on I with respect to µ. First, we are concerned with reducing the number of different coefficients in the three-term recurrence relation satisfied by these orthonormal rational functions. Next, we consider the case in which I = [– 1, 1] and µ satisfies the Erdos–Turán condition µ' > 0 a.e. on I (where µ' is the Radon–Nikodym derivative of the measure µ with respect to the Lebesgue measure) to discuss the convergence of n+1(x)/n(x) as n tends to infinity and to derive asymptotic formulas for the recurrence coefficients in the three-term recurrence relation. Finally, we give a strong convergence result for n(x) under the more restrictive condition that µ satisfies the Szeg
condition (1 – x2)–1/2 log µ'(x) 
L1([– 1, 1]).
Key Words: orthogonal rational functions; complex poles; three-term recurrence relation; asymptotics; ratio convergence; strong convergence