Asymptotic behaviour of Sobolev-type orthogonal polynomials on a rectifiable Jordan arc

Abstract

Our object of study is the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product $$\langle f, g \rangle = \int_{E} f(\xi) \overline{g(\xi)} \rho (\xi) \xi f(Z) A g(Z) ,$$ where $E$ is a rectifiable Jordan curve or arc in the complex plane $$f(Z) = (f(z_1), \ldots, f (l_1)}(z_1) , \ldots , f(z_m) , \ldots ,f (l_m)}(z_m)),$$ $A$ is an $M \times M$ Hermitian matrix, $M=l_{1} + \cdots + l_{m} + m$, $ \xi denotes the arc length measure, $\rho$ is a nonnegative function on $E$ , and $z_{i} \in \Omega$, $i=1,2,\ldots,m$, where $\Omega$ is the exterior region to $E$.