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Asymptotics for Gegenbauer-Sobolev type polynomials. (English) Zbl 0901.33001

Martínez Finkelshtein, Andrei (ed.) et al., Complex methods in approximation theory. Proceedings of the workshop, University of Almería, Spain, June 5–8, 1995. Almería: Univ. de Almería, Servicio de Publicaciones, Monografías Ciencia y Tecnología. 2, 85-91 (1997).
The authors investigate the asymptotic behaviour of the monic orthogonal polynomials \(Q^{(\alpha,\lambda, M)}_n\) with respect to the inner product \[ (f, g):= \langle f,g\rangle+ \lambda\langle f,g'\rangle+ M[f'(-1) g'(-1)+ f'(1)g'(1)], \] where \(\langle f,g\rangle:= \int^1_{-1} f(x)g(x)(1- x^2)^{\alpha- 1/2}dx\) with \(\alpha> -1/2\) and \(\lambda, M\geq 0\). Such inner products involving derivatives are called Sobolev-type products. The case \(\lambda= M= 0\) corresponds to the classical case of the Gegenbauer polynomials \(C^{(\alpha)}_n\).
The main result states that the asymptotic behaviour of the \(Q^{(\alpha,\lambda, 0)}_n\) is independent of \(\lambda> 0\); more precisely, for all \(\lambda> 0\), \[ {Q^{(\alpha,\lambda, 0)}_n(x)\over C^{(\alpha)}_n(x)}\to {1\over\phi'(x)}\qquad (n\to \infty), \] uniformly on compact subsets of \(\mathbb{C}\setminus[- 1,1]\), where \(\phi(x):={1\over 2} (x+ \sqrt{x^2- 1})\) with \(\sqrt{x^2- 1}> 0\) for \(x> 1\). A similar result is formulated for the general case \(M\geq 0\).
For the entire collection see [Zbl 0892.00034].
Reviewer: J.Müller (Trier)

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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