On a series representation for Carleman orthogonal polynomials

Abstract

Abstract. Let {p n (z)} ∞ n=0 be a sequence of complex polynomials (p n of degree n) that are orthonormal with respect to the area measure over the interior domain of an analytic Jordan curve. We prove that each p n of sufficiently large degree has a primitive that can be expanded in a series of functions recursively generated by a couple of integral transforms whose kernels are defined in terms of the degree n and the interior and exterior conformal maps associated with the curve. In particular, this series representation unifies and provides a new proof for two important known results: the classical theorem by Carleman establishing the strong asymptotic behavior of the polynomials p n in the exterior of the curve, and an integral representation that has played a key role in determining their behavior in the interior of the curve.

Asymptotics of Carleman Polynomials for Level Curves of the Inverse of a Shifted Zhukovsky Transformation

“…Carleman proved that (1) lim n→∞ p n (z) √ n + 1[φ(z)] n = φ (z), z ∈ Ω ρ , the convergence being uniform on compact subsets of Ω ρ (for a more complete statement, see [2, Satz IV], [5,Sec. 1], and also [3]). This establishes the asymptotic behavior of p n (z) on the closed exterior Ω 1 of L 1 , and on a portion of its interior G 1 , namely, on the "strip" Ω ρ ∩ G 1 .…”

“…A first version of Proposition 1.1 was proven in [10]. The version stated above is simpler to use and we shall briefly indicate at the end of Section 4 below how to derive it from the recent results of [3].…”

“…We finish this section by briefly indicating how to obtain Proposition 1.1 from the recent results of [3].…”

Asymptotics of polynomials orthogonal over circular multiply connected domains

“…The asymptotic properties of orthogonal polynomials over planar regions have been the focus of attention of past and many recent works. When the domain of orthogonality is bounded by a Jordan curve with some degree of smoothness (analytic, piecewise analytic, Hölder continuous, quasiconformal), strong asymptotics and/or zero distribution results have been derived in [1,2,3,4,5,11,12,13,19,23,24], and for orthogonality with weights, in [10,14,15]. Logarithmic/zero asymptotics with applications to shape reconstruction have been given in [8,17] for polynomials orthogonal over an archipelago (a finite union of Jordan domains).…”

On the leading coefficient of polynomials orthogonal over domains with corners

“…The asymptotic behavior as n → ∞ of these polynomials has been thoroughly investigated when L is an analytic Jordan curve in [1,2,3,4,7], while for L having some degree of smoothness, strong asymptotics for p n , outside and on the curve itself, were obtained by Suetin [10].…”