Zero distributions via orthogonality
Annales de l'Institut Fourier, Volume 55 (2005) no. 5, p. 1455-1499
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.
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.
DOI : https://doi.org/10.5802/aif.2130
Classification:  30C15,  30E10,  30E20,  31A15,  05E35,  42C05
Keywords: orthogonal polynomials, zero distribution, logarithmic potential, rational approximation
@article{AIF_2005__55_5_1455_0,
     author = {Baratchart, Laurent and K\"ustner, Reinhold and Totik, Vilmos},
     title = {Zero distributions via orthogonality},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {5},
     year = {2005},
     pages = {1455-1499},
     doi = {10.5802/aif.2130},
     mrnumber = {2172271},
     zbl = {1076.30010},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2005__55_5_1455_0}
}
Baratchart, Laurent; Küstner, Reinhold; Totik, Vilmos. Zero distributions via orthogonality. Annales de l'Institut Fourier, Volume 55 (2005) no. 5, pp. 1455-1499. doi : 10.5802/aif.2130. https://aif.centre-mersenne.org/item/AIF_2005__55_5_1455_0/

[1] J.-E. Andersson Best rational approximation to Markov functions, J. Approx. Theory, Tome 76 (1994) no. 2, pp. 219-232 | Article | MR 1268100 | Zbl 0796.41014

[2] L. Baratchart; V. Prokhorov; E.B. Saff Best meromorphic approximation of Markov functions on the unit circle, Found. Comput. Math., Tome 1 (2001) no. 4, pp. 385-416 | MR 1857722 | Zbl 1053.41019

[3] L. Baratchart; F. Seyfert An L p analog to AAK theory for p2, J. Funct. Anal., Tome 191 (2002) no. 1, pp. 52-122 | Article | MR 1909264 | Zbl 1050.47021

[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., Tome 105 (1999) no. 1-2 | MR 1690582 | Zbl 0967.30022

[5] L. Baratchart; H. Stahl; F. Wielonsky Asymptotic error estimates for L 2 best rational approximants to Markov functions, J. Approx. Theory, Tome 108 (2001) no. 1, pp. 53-96 | Article | MR 1808854 | Zbl 0973.41007

[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., Tome 17 (2001) no. 1, pp. 103-138 | MR 1794804 | Zbl 0983.30018

[7] L. Baratchart; F. Wielonsky Rational approximation in the real Hardy space H 2 and Stieltjes integrals: a uniqueness theorem, Constr. Approx., Tome 9 (1993) no. 1, pp. 1-21 | Article | MR 1198520 | Zbl 0769.30024

[8] R.A. Devore; G.G. Lorentz Constructive Approximation, Springer-Verlag, Berlin, Grundlehren der mathematischen Wissenschaften, Tome 303 (1993) | MR 1261635 | Zbl 0797.41016

[9] A.A. Gonchar; E.A. Rakhmanov Equilibrium distributions and degree of rational approximation of analytic functions, Math. USSR Sb., Tome 62 (1989) no. 2, pp. 305-348 | Article | MR 922628 | Zbl 0663.30039

[10] R. Kannan; C.K. Krueger Advanced Analysis on the Real Line, Springer-Verlag, New York, Universitext (1996) | Zbl 0855.26001

[11] H. Kestelman An integral for functions of bounded variation, J. London Math. Soc., Tome 9 (1934), pp. 174-178 | Article | Zbl 0009.30701

[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., Tome 19 (1987) no. 1, pp. 23-38 | MR 901209 | Zbl 0619.41014

[14] Q.I. Rahman; G. Schmeisser Analytic Theory of Polynomials, The Clarendon Press, Oxford (London Math. Soc. Monographs, New Series) Tome 26 (2002) | Zbl 01820648

[15] T. Ransford Potential Theory in the Complex Plane, Cambridge University Press, Cambridge (London Math. Soc. Student Texts) Tome 28 (1995) | Zbl 0828.31001

[16] E.B. Saff; V. Totik Logarithmic Potentials with External Fields, Springer-Verlag, Berlin, Grundlehren der mathematischen Wissenschaften, Tome 316 (1997) | MR 1485778 | Zbl 0881.31001

[17] H. Stahl The convergence of Padé approximants to functions with branch points, J. Approx. Theory, Tome 2 (1997) no. 91, pp. 139-204 | MR 1484040 | Zbl 0896.41009

[18] H. Stahl Orthogonal polynomials with respect to complex-valued measures, Orthogonal Polynomials and their Applications (Erice, 1990), Baltzer, Basel (IMACS Ann. Comput. Appl. Math.) Tome 9 (1990), pp. 139-154 | Zbl 0852.42009

[19] H. Stahl Orthogonal polynomial with complex-valued weight functions I, II, Constr. Approx., Tome 2 (1986) no. 3, p. 225-240, 241–251 | MR 891973 | Zbl 0592.42016

[20] H. Stahl; V. Totik General Orthogonal Polynomials, Cambridge University Press, Cambridge, Encyclopedia of Mathematics and its Applications, Tome 43 (1992) | MR 1163828 | Zbl 0791.33009

[21] J.L. Walsh Interpolation and Approximation by Rational Functions in the Complex Domain, third edition, Amer. Math. Soc., Providence Tome XX (1960) | MR 218587 | Zbl 0106.28104