Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials
- Series
- Analysis Seminar
- Time
- Tuesday, October 5, 2010 - 13:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 269
- Speaker
- Sasha Aptekarev – Keldish Institute for Applied Mathematics
The asymptotic theory is developed for polynomial sequences that
are generated by the three-term higher-order recurrence
Q_{n+1} = zQ_n - a_{n-p+1}Q_{n-p}, p \in \mathbb{N}, n\geq p,
where z is a complex variable and the coefficients a_k are
positive and satisfy the perturbation condition \sum_{n=1}^\infty
|a_n-a|<\infty . Our results generalize known results for p = 1,
that is, for orthogonal polynomial sequences on the real line that
belong to the Blumenthal-Nevai class. As is known, for p\geq 2,
the role of the interval is replaced by a starlike set S of
p+1 rays emanating from the origin on which the Q_n satisfy a
multiple orthogonality condition involving p measures. Here we
obtain strong asymptotics for the Q_n in the complex plane
outside the common support of these measures as well as on the
(finite) open rays of their support. In so doing, we obtain an
extension of Weyl's famous theorem dealing with compact
perturbations of bounded self-adjoint operators. Furthermore, we
derive generalizations of the classical Szeg\"o functions, and
we show that there is an underlying Nikishin system hierarchy for
the orthogonality measures that is related to the Weyl functions.
Our results also have application to Hermite-Pad\'e approximants
as well as to vector continued fractions.