An Expansion for Polynomials Orthogonal Over an Analytic Jordan Curve

Abstract

Abstract. We consider polynomials that are orthogonal over an analytic Jordan curve L with respect to a positive analytic weight, and show that each such polynomial of sufficiently large degree can be expanded in a series of certain integral transforms that converges uniformly in the whole complex plane. This expansion yields, in particular and simultaneously, Szegő's classical strong asymptotic formula and a new integral representation for the polynomials inside L. We further exploit such a representation to derive finer asymptotic results for weights having finitely many singularities (all of algebraic type) on a thin neighborhood of the orthogonality curve. Our results are a generalization of those previously obtained in [7] for the case of L being the unit circle. Introduction and statements of the resultsThe study of polynomials orthogonal over a closed rectifiable curve of the complex plane was initiated by Szegő in [20], and later continued by Szegő himself and such authors as Smirnov, Keldysh, Lavrentiev, Korovkin, Suetin and Geronimous (see [17] for references and an overview of the developments until 1964). Polynomials orthogonal over several arcs and curves have also been studied, for instance (and without being exhaustive), Among the central questions that are often investigated figure the asymptotic behavior of the orthogonal polynomials and the distribution and location of their zeros. In this regard, the case of a closed curve has the peculiarity (not observed in that of an open arc) that the interior of its polynomial convex hull is non-empty 1 , giving more freedom of distribution to the zeros of the polynomials, and consequently, making the behavior of the polynomials themselves much less clear. The results of the present work clarify this question to a substantial extent for a single closed curve under analyticity conditions. Let L 1 be an analytic Jordan curve in the complex plane C and let h(z) be an analytic function in a neighborhood of L 1 such that h(z) > 0 for all z ∈ L 1 . Using the Gram-Schmidt orthogonalization process, we can form a unique sequence {p n (z)} ∞ n=0 of orthonormal polynomials over L 1 with respect to h(z), i.e., satisfyingIn what follows, we are concerned with the asymptotic behavior of these polynomials as their degree n becomes large. With this generality, essentially the only known result is Szegő's strong asymptotic formulaHere φ is the conformal map of the exterior Ω 1 of L 1 onto the exterior of the unit circle satisfying that φ(∞) = ∞, φ ′ (∞) > 0, ∆ e (z; h) is the so-called exterior Szegő function for the weight h, and (3) holds locally uniformly as n → ∞ on any open set Ω ρ ⊃ Ω 1 that is conformally mapped by φ onto the exterior of a circle about the origin of radius ρ < 1, and is such that ∆ e (z; h) is analytic on Ω ρ (see next subsection for details). 1A well-known result by Widom [22] asserts that the zeros must accumulate, in the limit, on the polynomial convex hull of the support of the orthogonality measure. 2 ERWIN MIÑA-DÍAZ For h(z) ≡ 1, this formu...

On a series representation for Carleman orthogonal polynomials

“…Series representations similar to those established in this paper but for polynomials orthogonal over analytic Jordan curves were recently obtained in [5] and [6]. It is certainly worth exploring whether representations of this kind are also valid for other systems of orthogonal polynomials, in particular within the context of planar orthogonality, where the available tools for their asymptotic analysis are typically more limited.…”

Inverse Theorem on Row Sequences of Linear Padé-orthogonal Approximation

“…For general compact sets E contained in the complex plane the situation is not quite the same. There are many examples for which Szegő asymptotics takes place for measures supported on a single Jordan curve or arc (see [23,27,34,42,43,44]) and polynomials orthogonal with respect to area type measures on a Jordan region (see [12,26,28,33,35]). Outside of the previously mentioned cases of the segment and the unit circle, the only case fully described and easily verifiable where R(E) is substantially larger than S(E) is when E is an arc of the unit circle, see [3,Theorem 1] and [6,Theorem 1].…”

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“…(we identify ν (e iθ ) and ν (θ)). As in [14], define ∆ r,q (z) by ∆ r,q (z) = exp 1 2qπi Γr log (w(ζ))…”