An Asymptotic Integral Representation for Carleman Orthogonal Polynomials

Miña-Díaz, Erwin

doi:10.1093/imrn/rnn065

Cited by 16 publications

(27 citation statements)

References 14 publications

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“…1 There is a well developed theory of Bergman polynomials -their basic properties, and the asymptotic behavior, including that of their zeros [6], [7], [8], [12], [18], [19], [23]. In describing these, the conformal map of the exterior of , namely D = Cn G onto the exterior of the unit ball plays a key role.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Universality Type Limits for Bergman Orthogonal Polynomials

Lubinsky

2010

Comput. Methods Funct. Theory

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Section: Introduction and Resultsmentioning

confidence: 99%

Universality Type Limits for Bergman Orthogonal Polynomials

Lubinsky

2010

Comput. Methods Funct. Theory

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“…For more details, see Corollary 1.4 below and Theorem 2.1.2 of [7], where this representation was originally obtained and applied to derive precise asymptotics for p n in G 1 under some additional geometric considerations.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Stated as above, Corollary 1.4 suffices to obtain precise asymptotics for P n (z) in G 1 under the geometric assumption that (roughly speaking) ∂Ω ρ is a piecewise analytic curve (see [7]). To derive more general results, the following improvement is needed:…”

Section: Corollary 13 (14)mentioning

confidence: 99%

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On a series representation for Carleman orthogonal polynomials

Dragnev

Miña-Díaz

2010

Proc. Amer. Math. Soc.

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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.

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“…For instance, if h(z) ≡ 1, then ∆ e (z) ≡ 1, ∆ i (z) ≡ 1, and the behavior of P n inside L 1 only depends on geometric considerations and can be determined with great precision, for instance, when ∂Ω b ρ is a piecewise analytic curve, in which case the map ψ has finitely many singularities on the circle T b ρ , having an asymptotic expansion about each of them. We will not pursue the analysis of this case here as it is very similar to the one already carried out in [8] for polynomials orthogonal over the interior of an analytic curve with respect to area measure. Instead, we shall concentrate on a case where the behavior of P n is only influenced by the singularities of ∆ e (z), which are finitely many, all lying on the band G 1 ∩ Ω b ρ and of algebraic type.…”

Section: The Expansionsmentioning

confidence: 99%

An Expansion for Polynomials Orthogonal Over an Analytic Jordan Curve

Miña-Díaz

2008

Commun. Math. Phys.

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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...

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An Asymptotic Integral Representation for Carleman Orthogonal Polynomials

Cited by 16 publications

References 14 publications

Universality Type Limits for Bergman Orthogonal Polynomials

Universality Type Limits for Bergman Orthogonal Polynomials

On a series representation for Carleman orthogonal polynomials

An Expansion for Polynomials Orthogonal Over an Analytic Jordan Curve

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