Orthogonal polynomials with complex-valued weight function, I

Abstract

In this paper we investigate the asymptotic behavior of polynomialsQ _mn(z), m, n ∈ N, of degree ≤n that satisfy the orthogonal relation

$$\oint_c {\zeta ^l Q_{mn} (\zeta )} \frac{{f(\zeta )d\zeta }}{{\omega _{m + n} (\zeta )}} = 0,l = 0,...,n - 1,$$

where/tf(z) is a function, which is supposed to be analytic on a continuum$V \subseteq \hat C$ and all its singularities are supposed to be contained in a set$E \subseteq \hat C$ of capacity zero, ω _m+n (z) is a polynomial of degreem+n+1 with all its zeros contained inV, andC is a curve separatingV from the setE.

We show that if the zeros of ω _m+n have a certain asymptotic distribution form+n → ∞ and ifm/n ar 1, then the zeros of the polynomialsQ _mn have a unique asymptotic distribution, which is closely related with the extremal domainD for single-valued analytic continuation of the functionf(z). The results are essential for the investigation of Padé and best rational approximants to the functionf(z).