A Riemann-Hilbert representation for Sobolev orthogonal polynomials

Abstract

In this article we consider polynomials orthogonal with respect to the inner product $$⟨P,Q{⟩}_S={\int }_{\mathbb{R}}P(x)Q(x){\mathrm{e}}^{-V(x)}\mathrm{d}x+\lambda {\int }_{\mathbb{R}}{P}^{\prime }(x)Q^{\prime }(x){\mathrm{e}}^{-V(x)}\mathrm{d}x$$ where $V$ is a polynomial of even degree (at least four) and positive leading coefficient, and $\lambda >0$ a constant. The above inner product is a special case of the so-called Sobolev inner product. We show how one can represent the associated Sobolev orthogonal polynomials as a species of Type I multiple-orthogonal polynomial. This correspondence makes use of the WKB asymptotics of a certain second order ODE. From this we may write a Riemann--Hilbert problem for the Sobolev orthogonal polynomials, from which one can deduce a Christoffel--Darboux-type formula for the projection kernel.

31 pages