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Asymptotics of Sobolev orthogonal polynomials for Hermite coherent pairs. (English) Zbl 0990.42011
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 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.

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Full Text: DOI
[1] Marcellán, F.; Martı́nez-Finkelshtein, A.; Moreno-Balcázar, J.J., Asymptotics of Sobolev orthogonal polynomials for symmetrically coherents pairs of measures with compact support, J. comput. appl. math., 81, 211-216, (1997)
[2] F. Marcellán, J.J. Moreno-Balcázar, Strong and Plancherel-Rotach asymptotics of non-diagonal Laguerre-Sobolev orthogonal polynomials, J. Approx. Theory, to be published.
[3] Martı́nez-Finkelshtein, A.; Moreno-Balcázar, J.J.; Pérez, T.E.; Piñar, M.A., Asymptotics of Sobolev orthogonal polynomials for coherent pairs, J. approx. theory, 92, 280-293, (1998) · Zbl 0898.42006
[4] Martı́nez-Finkelshtein, A., Bernstein-szegő’s theorem for Sobolev orthogonal polynomials, Constr. approx., 16, 73-84, (2000) · Zbl 0951.42014
[5] Meijer, H.G., Determination of all coherent pairs, J. approx. theory, 89, 321-343, (1997) · Zbl 0880.42012
[6] Meijer, H.G.; Pérez, T.E.; Piñar, M.A., Asymptotics of Sobolev orthogonal polynomials for coherent pairs of Laguerre type, J. math. anal. appl., 245, 528-546, (2000) · Zbl 0965.42017
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[8] Van Assche, W., Asymptotics for orthogonal polynomials, lecture notes in mathematics, vol. 1265, (1987), Springer Berlin
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