Relative asymptotics of orthogonal polynomials for perturbed measures

E. B. Saff   ; N. Stylianopoulos

Sbornik. Mathematics, Tome 209 (2018) no. 3, pp. 449-468

Résumé

We survey and present some new results that are related to the behaviour of orthogonal polynomials in the plane under small perturbations of the measure of orthogonality. More precisely, we introduce the notion of a polynomially small (PS) perturbation of a measure. Namely, if $\mu_0 \geqslant \mu_1$ and $\{p_n(\mu_j,z)\}_{n=0}^\infty$, $j=0,1$, are the associated orthonormal polynomial sequences, then $\mu_0$ is a PS perturbation of $\mu_1$ if $\|p_n(\mu_1,\,\cdot\,)\|_{L_2(\mu_0-\mu_1)}\to 0$, as $n\to\infty$. In such a case we establish relative asymptotic results for the two sequences of orthonormal polynomials. We also provide results dealing with the behaviour of the zeros of PS perturbations of area orthogonal (Bergman) polynomials. Bibliography: 35 titles.

Détail

Keywords: Christoffel function, Bergman polynomial, perturbed measure.
Mots-clés : orthogonal polynomial

Bibliographie

[1] A. Ambroladze, “On exceptional sets of asymptotic relations for general orthogonal polynomials”, J. Approx. Theory, 82:2 (1995), 257–273 | DOI | MR | Zbl

[2] N. Bosuwan, G. López Lagomasino, E. B. Saff, “Determining singularities using rows of Padé-orthogonal approximants”, Jaen J. Approx., 5:2 (2013), 179–208 | MR | Zbl

[3] D. Gaier, Lektsii po teorii approksimatsii v kompleksnoi oblasti, Mir, M., 1986, 216 pp. ; D. Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser Verlag, Basel–Boston, Mass., 1980, 174 pp. ; D. Gaier, Lectures on complex approximation, Birkhäuser Boston Inc., Boston, MA, 1987, xvi+196 с. | MR | Zbl | MR | Zbl | DOI | MR | Zbl

[4] B. Gustafsson, M. Putinar, E. B. Saff, N. Stylianopoulos, “Bergman polynomials on an archipelago: estimates, zeros and shape reconstruction”, Adv. Math., 222:4 (2009), 1405–1460 | DOI | MR | Zbl

[5] D. Khavinson, N. Stylianopoulos, “Recurrence relations for orthogonal polynomials and algebraicity of solutions of the Dirichlet problem”, Around the research of Vladimir Maz'ya, v. II, Int. Math. Ser. (N. Y.), 12, Springer, New York, 2010, 219–228 | DOI | MR | Zbl

[6] H. N. Mhaskar, E. B. Saff, “The distribution of zeros of asymptotically extremal polynomials”, J. Approx. Theory, 65:3 (1991), 279–300 | DOI | MR | Zbl

[7] E. Miña-Díaz, “Asymptotics of polynomials orthogonal over the unit disk with respect to a polynomial weight without zeros on the unit circle”, J. Approx. Theory, 165:1 (2013), 41–59 | DOI | MR | Zbl

[8] E. Miña-Díaz, E. B. Saff, N. S. Stylianopoulos, “Zero distributions for polynomials orthogonal with weights over certain planar regions”, Comput. Methods Funct. Theory, 5:1 (2005), 185–221 | DOI | MR | Zbl

[9] F. Nazarov, A. Volberg, P. Yuditskii, “Asymptotics of orthogonal polynomials via the Koosis theorem”, Math. Res. Lett., 13:5-6 (2006), 975–983 | DOI | MR | Zbl

[10] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren Math. Wiss., 299, Springer-Verlag, Berlin, 1992, x+300 pp. | DOI | MR | Zbl

[11] I. E. Pritsker, “Approximation of conformal mapping via the Szegő kernel method”, Comput. Methods Funct. Theory, 3:1–2 (2003), 79–94 | DOI | MR | Zbl

[12] M. Putinar, N. Stylianopoulos, “Finite-term relations for planar orthogonal polynomials”, Complex Anal. Oper. Theory, 1:3 (2007), 447–456 | DOI | MR | Zbl

[13] T. Ransford, Potential theory in the complex plane, London Math. Soc. Stud. Texts, 28, Cambridge Univ. Press, Cambridge, 1995, x+232 pp. | DOI | MR | Zbl

[14] E. B. Saff, “Orthogonal polynomials from a complex perspective”, Orthogonal polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 294, Kluwer Acad. Publ., Dordrecht, 1990, 363–393 | DOI | MR | Zbl

[15] E. B. Saff, “Remarks on relative asymptotics for general orthogonal polynomials”, Recent trends in orthogonal polynomials and approximation theory, Contemp. Math., 507, Amer. Math. Soc., Providence, RI, 2010, 233–239 | DOI | MR | Zbl

[16] E. B. Saff, N. Stylianopoulos, “Asymptotics for Hessenberg matrices for the Bergman shift operator on Jordan regions”, Complex Anal. Oper. Theory, 8:1 (2014), 1–24 | DOI | MR | Zbl

[17] E. B. Saff, N. Stylianopoulos, “On the zeros of asymptotically extremal polynomial sequences in the plane”, J. Approx. Theory, 191 (2015), 118–127 | DOI | MR | Zbl

[18] E. B. Saff, Computational complex analysis and approximation theory (Protaras, Cyprus, 2011), 2011, Invited talk in the conference

[19] E. B. Saff, V. Totik, Logarithmic potentials with external fields, Grundlehren Math. Wiss., 316, Springer-Verlag, Berlin, 1997, xvi+505 pp. | DOI | MR | Zbl

[20] E. B. Saff, H. Stahl, N. Stylianopoulos, V. Totik, “Orthogonal polynomials for area-type measures and image recovery”, SIAM J. Math. Anal., 47:3 (2015), 2442–2463 | DOI | MR | Zbl

[21] B. Simanek, “A new approach to ratio asymptotics for orthogonal polynomials”, J. Spectr. Theory, 2:4 (2012), 373–395 | DOI | MR | Zbl

[22] B. Simanek, “Asymptotic properties of extremal polynomials corresponding to measures supported on analytic regions”, J. Approx. Theory, 170 (2013), 172–197 | DOI | MR | Zbl

[23] B. Simanek, “Ratio asymptotics, Hessenberg matrices, and weak asymptotic measures”, Int. Math. Res. Not. IMRN, 2014:24 (2014), 6798–6825 | DOI | MR | Zbl

[24] B. Simon, Orthogonal polynomials on the unit circle, Part 1. Classical theory, Amer. Math. Soc. Colloq. Publ., 54, Part 1, Amer. Math. Soc., Providence, RI, 2005, xxvi+466 pp. | MR | Zbl

[25] H. Stahl, V. Totik, “$N$th root asymptotic behavior of orthonormal polynomials”, Orthogonal polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 294, Kluwer Acad. Publ., Dordrecht, 1990, 395–417 | DOI | MR | Zbl

[26] H. Stahl, V. Totik, General orthogonal polynomials, Encyclopedia Math. Appl., 43, Cambridge Univ. Press, Cambridge, 1992, xii+250 pp. | DOI | MR | Zbl

[27] N. Stylianopoulos, “Strong asymptotics for Bergman polynomials over domains with corners and applications”, Constr. Approx., 38:1 (2013), 59–100 | DOI | MR | Zbl

[28] N. Stylianopoulos, “Boundary estimates for Bergman polynomials in domains with corners”, Modern trends in constructive function theory, Contemp. Math., 661, Amer. Math. Soc., Providence, RI, 2016, 187–198 | DOI | MR | Zbl

[29] P. K. Suetin, “Fundamental properties of polynomials orthogonal on a contour”, Russian Math. Surveys, 21:2 (1966), 35–83 | DOI | MR | Zbl

[30] P. K. Suetin, “Polynomials orthogonal over a region and Bieberbach polynomials”, Proc. Steklov Inst. Math., 100 (1974), 1–91 | MR | Zbl

[31] V. Totik, “Orthogonal polynomials”, Surv. Approx. Theory, 1 (2005), 70–125 | MR | Zbl

[32] V. Totik, “Christoffel functions on curves and domains”, Trans. Amer. Math. Soc., 362:4 (2010), 2053–2087 | DOI | MR | Zbl

[33] V. Totik, T. Varga, “Chebyshev and fast decreasing polynomials”, Proc. Lond. Math. Soc. (3), 110:5 (2015), 1057–1098 | DOI | MR | Zbl

[34] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Colloq. Publ., XX, 4th ed., Amer. Math. Soc., Providence, RI, 1965, x+405 pp. | MR | MR | Zbl | Zbl

[35] H. Widom, “Polynomials associated with measures in the complex plane”, J. Math. Mech., 16 (1967), 997–1013 | MR | Zbl