In this paper we investigate the asymptotic behavior of polynomials that are orthonormal over the interior domain of an analytic Jordan curve L with respect to area measure. We prove that, inside L, these polynomials behave asymptotically like a sequence of certain integrals involving the canonical conformal map of the exterior of L onto the exterior of the unit circle and certain meromorphic kernel function defined in terms of a conformal map of the interior of L onto the unit disk. The error term in the integral representation is proven to decay geometrically and sufficiently fast, allowing us to obtain more precise asymptotic formulas for the polynomials under certain additional geometric considerations. These formulas yield, in turn, fine results on the location, limiting distribution and accumulation points of the zeros of the polynomials.
Polynomials Qn(z), n = 0, 1, . . . , that are multi-orthogonal with respect to a Nikishin system of p ≥ 1 compactly supported measures over the star-like set of p + 1 rays S+ := {z ∈ C : z p+1 ≥ 0} are investigated. We prove that the Nikishin system is normal, that the polynomials satisfy a three-term recurrence relation of order p + 1 of the form zQn(z) = Qn+1(z) + an Qn−p(z) with an > 0 for all n ≥ p, and that the nonzero roots of Qn are all simple and located in S+. Under the assumption of regularity (in the sense of Stahl and Totik) of the measures generating the Nikishin system, we describe the asymptotic zero distribution and weak behavior of the polynomials Qn in terms of a vector equilibrium problem for logarithmic potentials. Under the same regularity assumptions, a theorem on the convergence of the Hermite-Padé approximants to the Nikishin system of Cauchy transforms is proven.
Let L be an analytic Jordan curve and let { p n (z)} ∞ n=0 be the sequence of polynomials that are orthonormal with respect to the area measure over the interior of L. A well-known result of Carleman states that lim n→∞ p n (z)locally uniformly on a certain open neighborhood of the closed exterior of L, where φ is the canonical conformal map of the exterior of L onto the exterior of the unit circle. In this paper we extend the validity of (1) to a maximal open set, every boundary point of which is an accumulation point of the zeros of the p n 's. Some consequences on the limiting distribution of the zeros are discussed, and the results are illustrated with two concrete examples and numerical computations.
The sine process is a rigid point process on the real line, which means that for almost all configurations X, the number of points in an interval I = [−R, R] is determined by the points of X outside of I. In addition, the points in I are an orthogonal polynomial ensemble on I with a weight function that is determined by the points in X \I. We prove a universality result that in particular implies that the correlation kernel of the orthogonal polynomial ensemble tends to the sine kernel as the length |I| = 2R tends to infinity, thereby answering a question posed by A.I. Bufetov.
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