Zero distributions via orthogonality
[Distributions asymptotiques de zéros et orthogonalité]
Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1455-1499.

On développe une nouvelle méthode pour établir la distribution asymptotique des zéros de divers polynômes orthogonaux sur un segment. Cette méthode exploite de manière directe les relations d'orthogonalité. Nous l'illustrons dans quatre cas : l'orthogonalité classique par rapport à une mesure positive, l'orthogonalité non-Hermitienne par rapport à une mesure complexe, et l'orthogonalité non-linéaire intervenant en approximation rationnelle, tout d'abord dans le cas d'une mesure positive, puis dans le cas non- Hermitien.

We develop a new method to prove asymptotic zero distribution for different kinds of orthogonal polynomials. The method directly uses the orthogonality relations. We illustrate the procedure in four cases: classical orthogonality, non-Hermitian orthogonality, orthogonality in rational approximation of Markov functions and its non- Hermitian variant.

DOI : 10.5802/aif.2130
Classification : 30C15, 30E10, 30E20, 31A15, 05E35, 42C05
Keywords: orthogonal polynomials, zero distribution, logarithmic potential, rational approximation
Mot clés : polynômes orthogonaux, distribution des zèros, potentiel logarithmique, approximation rationnelle
Baratchart, Laurent 1 ; Küstner, Reinhold  ; Totik, Vilmos 

1 INRIA, 2004 route des Lucioles, BP 93, 06902 Sophia-Antipolis Cedex (France), Université de Provence, LATP, CMI, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13 (France), University of Szeged, Bolyai Institute, Aradi v. tere 1, 6720 (Hongrie), University of South Florida, department of mathematics, Tampa FL 33620 (USA)
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Baratchart, Laurent; Küstner, Reinhold; Totik, Vilmos. Zero distributions via orthogonality. Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1455-1499. doi : 10.5802/aif.2130. https://aif.centre-mersenne.org/articles/10.5802/aif.2130/

[1] J.-E. Andersson Best rational approximation to Markov functions, J. Approx. Theory, Volume 76 (1994) no. 2, pp. 219-232 | DOI | MR | Zbl

[2] L. Baratchart; V. Prokhorov; E.B. Saff Best meromorphic approximation of Markov functions on the unit circle, Found. Comput. Math., Volume 1 (2001) no. 4, pp. 385-416 | MR | Zbl

[3] L. Baratchart; F. Seyfert An L p analog to AAK theory for p2, J. Funct. Anal., Volume 191 (2002) no. 1, pp. 52-122 | DOI | MR | Zbl

[4] L. Baratchart; H. Stahl; F. Wielonsky Non-uniqueness of rational best approximants, Continued Fractions and Geometric Function Theory (CONFUN), (Trondheim, 1997), J. Comput. Appl. Math., Volume 105 (1999) no. 1-2 | MR | Zbl

[5] L. Baratchart; H. Stahl; F. Wielonsky Asymptotic error estimates for L 2 best rational approximants to Markov functions, J. Approx. Theory, Volume 108 (2001) no. 1, pp. 53-96 | DOI | MR | Zbl

[6] L. Baratchart; H. Stahl; F. Wielonsky Asymptotic uniqueness of best rational approximants of given degree to Markov functions in L 2 of the circle, Constr. Approx., Volume 17 (2001) no. 1, pp. 103-138 | MR | Zbl

[7] L. Baratchart; F. Wielonsky Rational approximation in the real Hardy space H 2 and Stieltjes integrals: a uniqueness theorem, Constr. Approx., Volume 9 (1993) no. 1, pp. 1-21 | DOI | MR | Zbl

[8] R.A. DeVore; G.G. Lorentz Constructive Approximation, Grundlehren der mathematischen Wissenschaften, 303, Springer-Verlag, Berlin, 1993 | MR | Zbl

[9] A.A. Gonchar; E.A. Rakhmanov Equilibrium distributions and degree of rational approximation of analytic functions, Math. USSR Sb., Volume 62 (1989) no. 2, pp. 305-348 | DOI | MR | Zbl

[10] R. Kannan; C.K. Krueger Advanced Analysis on the Real Line, Universitext, Springer-Verlag, New York, 1996 | Zbl

[11] H. Kestelman An integral for functions of bounded variation, J. London Math. Soc., Volume 9 (1934), pp. 174-178 | DOI | Zbl

[12] R. Küstner Asymptotic zero distribution of orthogonal polynomials with respect to complex measures having argument of bounded variation (2003) (Ph.D. thesis, University of Nice Sophia Antipolis, http://www.inria.fr/rrrt/tu-0784.html)

[13] A. Magnus Toeplitz matrix techniques and convergence of complex weight Padé approximants, J. Comput. Appl. Math., Volume 19 (1987) no. 1, pp. 23-38 | MR | Zbl

[14] Q.I. Rahman; G. Schmeisser Analytic Theory of Polynomials (London Math. Soc. Monographs, New Series), Volume 26 (2002) | Zbl

[15] T. Ransford Potential Theory in the Complex Plane (London Math. Soc. Student Texts), Volume 28 (1995) | Zbl

[16] E.B. Saff; V. Totik Logarithmic Potentials with External Fields, Grundlehren der mathematischen Wissenschaften, 316, Springer-Verlag, Berlin, 1997 | MR | Zbl

[17] H. Stahl The convergence of Padé approximants to functions with branch points, J. Approx. Theory, Volume 2 (1997) no. 91, pp. 139-204 | MR | Zbl

[18] H. Stahl Orthogonal polynomials with respect to complex-valued measures, Orthogonal Polynomials and their Applications (Erice, 1990) (IMACS Ann. Comput. Appl. Math.), Volume 9 (1990), pp. 139-154 | Zbl

[19] H. Stahl Orthogonal polynomial with complex-valued weight functions I, II, Constr. Approx., Volume 2 (1986) no. 3, p. 225-240, 241–251 | MR | Zbl

[20] H. Stahl; V. Totik General Orthogonal Polynomials, Encyclopedia of Mathematics and its Applications, 43, Cambridge University Press, Cambridge, 1992 | MR | Zbl

[21] J.L. Walsh Interpolation and Approximation by Rational Functions in the Complex Domain, third edition, XX, Amer. Math. Soc., Providence, 1960 | MR | Zbl

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