Ratio asymptotics for orthogonal rational functions on an interval

• Authors: J. Van Deun;A. Bultheel
• Venue: Journal of Approximation Theory
• Year: 2003

Let {α1, α2,...} be a sequence of real numbers outside the interval [-1, 1] and µ a positive bounded Borel measure on this interval satisfying the Erdös-Turán condition µ' 0 a.e., where µ' is the Radon-Nikodym derivative of the measure µ with respect to the Lebesgue measure. We introduce rational functions φn(x) with poles {α1, ..., αn} orthogonal on [-1, 1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of φn-1(x)/φn(x) as n tends to infinity under certain assumptions on the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three-term recurrence relation satisfied by the orthonormal functions.