Strong asymptotics for Sobolev orthogonal polynomials in the complex plane

Abstract

We obtain the strong asymptotics for the sequence of monic polynomials minimizing the norm $$\Vert q_S\Vert=(\sum_{k=0} \Vert q (k)}\Vert_k ) 1/2}$$ where $\Vert \bullet\Vert_k$, $k=0,\dots, N-1$, are $L $ norms with respect to measures supported on the same rectifiable Jordan closed curve on $\Gamma$, and $\Vert \bullet\Vert_N$ is the $L $ norm corresponding to a weight supported or arc $\Gamma$, which satisfies the Szegö condition, plus mass points in the unbounded connected component of $\bbfC\setminus \Gamma$.