Asymptotics for Sobolev orthogonal polynomials for exponential weights

Abstract

^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.