Asymptotics for Sobolev orthogonal polynomials for exponential weights

38 pages, no figures.-- MSC2000 codes: 42C05, 33C25.MR#: MR2164139 (2006c:41040)Zbl#: Zbl 1105.42016^aLet $\lambda >0,\alpha >1$, and let $W( x) =\exp ( -\vert x\vert ^{\alpha })$, x\in \mbox{\smallbf R}. Let \psi \in L_{\infty }(\mbox{\smallbf R}) be positive on a set of positive measure. For $n\geq 1$, one may form Sobolev orthonormal polynomials $( q_{n})$, associated with the Sobolev inner product ( f,g) =\int_{\mbox{\scriptsize\bf R}}fg( \psi W) ^{2}+\lambda \int_{\mbox{\scriptsize\bf R}}f^{\prime }g^{\prime }W^{2}. We establish strong asymptotics for the $( q_{n})$ in terms of the ordinary orthonormal polynomials $( p_{n})$ for the weight $W^{2}$, on and off the real line. More generally, we establish a close asymptotic relationship between $( p_{n})$ and $( q_{n})$ for exponential weights $W=\exp ( -Q)$ on a real interval $I$, under mild conditions on $Q$. The method is new and will apply to many situations beyond that treated in this paper.The work by F. Marcellan has been supported by Dirección General de Investigación (Ministerio de Ciencia y Technología) of Spain under grant BFM 2003-06335-C03-07, as well as NATO Collaborative grant PST.CLG 979738. J. Geronimo and D. Lubinsky, respectively, acknowledge support by NSF grants DMS-0200219 and DMS-0400446.Publicad

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