an:01659263
Zbl 0990.42011
Alfaro, Manuel; Moreno-Balc??zar, Juan J.; P??rez, Teresa E.; Pi??ar, Miguel A.; Rezola, M. Luisa
Asymptotics of Sobolev orthogonal polynomials for Hermite coherent pairs
EN
J. Comput. Appl. Math. 133, No. 1-2, 141-150 (2001).
00077409
2001
j
42C05 33C45
Sobolev orthogonal polynomials; asymptotics; symmetrically coherent pairs; Hermite polynomials; Plancherel-Rotach asymptotics
A wide range of results regarding algebraic and analytic properties of polynomials (say, \(Q_n\)), orthogonal with respect to an inner product of the form
\[
(f,g)_S=\int f g d\mu_0+\int f' g' d\mu_1
\]
is obtained under additional assumption that the measures \(\mu_0\) and \(\mu_1\) form a so-called coherent pair. If supported on the whole \(\mathbb R\), either one of the measures \(\mu_k\) from the coherent pair is \(\exp(-x^2) dx\), and the corresponding sequence of monic Sobolev orthogonal polynomials form a one-parametric family, fully described by \textit{H. G. Meijer} [J. Approximation Theory 89, No. 3, 321-343 (1997; Zbl 0880.42012)].
In this setting the authors prove several asymptotic results for \(Q_n\) (as \(n \to \infty\)). First, they establish the behavior of \(Q_n/H_n\) (where \(H_n\) are the Hermite polynomials) in \(\mathbb C \setminus \mathbb R\). Further, they describe the behavior of this fraction with scaled variable, from where a Plancherel-Rotach asymptotics and the accumulation set of scaled zeros for \(Q_n\) follow.
Andrei Mart??nez Finkelshtein (Almeria)
Zbl 0880.42012