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

Abstract

This paper complements the recent investigation of [4] on the asymptotic behavior of polynomials orthogonal over the interior of an analytic Jordan curve L. We study the specific case of L = {z = w−1+(w−1) −1 , |w| = R}, for some R > 2, providing an example that exhibits the new features discovered in [4], and for which the asymptotic behavior of the orthogonal polynomials is established over the entire domain of orthogonality. Surprisingly, this variation of the classical example of the ellipse turns out to be quite sophisticated. After properly normalizing the corresponding orthonormal polynomials pn, n = 0, 1, . . ., and on certain critical subregion of the orthogonality domain, a subsequence {pn k } converges if and only if log µ 4 (n k ) converges modulo 1 (µ being an important quantity associated to L). As a consequence, the limiting points of the sequence {pn} form a one parameter family of functions, the parameter's range being the interval [0, 1). The polynomials pn are much influenced by a certain integrand function, the explained behavior being the result of this integrand having a nonisolated singularity that is a cluster point of poles. The nature of this singularity sparks purely from geometric considerations, as opposed to the more common situation where the critical singularities come from the orthogonality weight.1991 Mathematics Subject Classification. 42C05, 30E10, 30E15.

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).…”

“…Let (n k ) be a subsequence of the natural numbers and let σ ∈ (0, 1). As proven in [5,Theorem 2], the sequence of functions (Θσ(n k t)) ∞ k=1 converges normally on ℜt < 0 if and only if there exists q ∈ [0, 1) such that log σ n k → q modulo 1, that is,…”

“…Our job here is more of translating what was accomplished there into our current setting. We will therefore lay out the main steps involved in proving (5.5), indicating in each case where to find the full explanation in [6]. Let Bj be the disk defined by (2.7), and note that Φj (D(0, |aj |)) = Bj .…”

“…which is valid for every pair (m, n) ∈ N × N and for every t ∈ C \ (−∞, −|aj| −1 ]. This representation can be derived by using the summation by parts formula in conjunction with (6.25); for the details see Lemma 3.3 in [6] and its proof. From the definition (6.22), we find 4 , so that, as n → ∞,…”

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].…”