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Do Orthogonal Polynomials Dream of Symmetric Curves?

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Abstract

The complex or non-Hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not clear. Nevertheless, numerical experiments, such as those presented in this paper, show that the zeros not simply cluster somewhere on the plane, but persistently choose to align on certain curves, and in a very regular fashion. The problem of the limit zero distribution for the non-Hermitian orthogonal polynomials is one of the central aspects of their theory. Several important results in this direction have been obtained, especially in the last 30 years, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation. Thus, the second motivation of this paper is to discuss some “mysterious” configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. In this apparently simple situation the zeros of these orthogonal polynomials may exhibit different behaviors: for some of them we state the rigorous results, while others are presented as conjectures (apparently, within a reach of modern techniques). Finally, there are cases for which it is not yet clear how to explain our numerical results, and where we cannot go beyond an empirical discussion.

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Notes

  1. The procedure used to compute these zeros numerically is briefly explained in Sect. 6.2.

  2. As a referee pointed out, the configuration of Fig. 7, top right, could be formally considered as a limit case of what we describe in Theorem 6.1 below, where the S-property does play a role. Nevertheless, the situation is completely different here due to the existence of a string of zeros of the weight (6.6) along the imaginary axis. In fact, zeros of \(Q_n\), at least visually, are equally spaced on \(i{\mathbb {R}}\), following the distribution of zeros of \(w_n\) and not an equilibrium measure on an interval of \(i{\mathbb {R}}\).

  3. Strictly speaking, even the situation when the components of the support of \(\vec {\lambda }\) are simple arcs with common endpoints is already interesting.

References

  1. G. Álvarez, L. Martínez Alonso, and E. Medina. Determination of \(S\)-curves with applications to the theory of non-Hermitian orthogonal polynomials. J. Stat. Mech. Theory Exp., (6):P06006, 28, 2013.

  2. G. Álvarez, L. Martínez Alonso, and E. Medina. Partition functions and the continuum limit in Penner matrix models. J. Phys. A, 47(31):315205, 29, 2014.

  3. G. Álvarez, L. Martínez Alonso, and E. Medina. Fine structure in the large \(n\) limit of the non-Hermitian Penner matrix model. Ann. Physics, 361:440–460, 2015.

    Article  MathSciNet  Google Scholar 

  4. A. I. Aptekarev. Strong asymptotics of polynomials of simultaneous orthogonality for Nikishin systems. Mat. Sb., 190(5):3–44, 1999.

    Article  MathSciNet  Google Scholar 

  5. A. I. Aptekarev. Asymptotics of Hermite-Padé approximants for a pair of functions with branch points. Dokl. Akad. Nauk, 422(4):443–445, 2008.

    MathSciNet  Google Scholar 

  6. A. I. Aptekarev, V. I. Buslaev, A. Martines-Finkel shteĭn, and S. P. Suetin. Padé approximants, continued fractions, and orthogonal polynomials. Uspekhi Mat. Nauk, 66(6(402)):37–122, 2011.

  7. A. I. Aptekarev and A. B. È. Koĭèlaars. Hermite-Padé approximations and ensembles of multiple orthogonal polynomials. Uspekhi Mat. Nauk, 66(6(402)):123–190, 2011.

  8. A. I. Aptekarev, G. Lopes Lagomasino, and I. A. Rocha. Asymptotic behavior of the ratio of Hermite-Padé polynomials for Nikishin systems. Mat. Sb., 196(8):3–20, 2005.

    Article  MathSciNet  Google Scholar 

  9. A. I. Aptekarev and H. Stahl. Asymptotics of Hermite-Padé polynomials. In Progress in approximation theory (Tampa, FL, 1990), volume 19 of Springer Ser. Comput. Math., pages 127–167. Springer, New York, 1992.

  10. A. I. Aptekarev and M. L. Yattselev. Padé approximants for functions with branch points – strong asymptotics of Nuttall-Stahl polynomials. arXiv:1109.0332, 2011.

  11. J. Baik, T. Kriecherbauer, K. D. T.-R. McLaughlin, and P. D. Miller. Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles: announcement of results. Int. Math. Res. Not., (15):821–858, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin, and P. D. Miller. Discrete orthogonal polynomials, volume 164 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2007. Asymptotics and applications.

  13. George A. Baker, Jr. and P. Graves-Morris. Padé approximants, volume 59 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 1996.

  14. F. Balogh, M. Bertola, and T. Bothner. Hankel determinant approach to generalized Vorob’ev-Yablonski polynomials and their roots. Preprint arXiv:1504.00440.

  15. L. Baratchart, R. Küstner, and V. Totik. Zero distributions via orthogonality. Ann. Inst. Fourier (Grenoble), 55(5):1455–1499, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Baratchart, H. Stahl, and M. Yattselev. Weighted extremal domains and best rational approximation. Adv. Math., 229(1):357–407, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Baratchart, and M. Yattselev. Convergent interpolation to Cauchy integrals over analytic arcs with Jacobi-type weights. Int. Math. Res. Not. IMRN no. 22, 4211–4275, 2010.

  18. M. Bertola and T. Bothner. Zeros of large degree Vorob’ev-Yablonski polynomials via a Hankel determinant identity. Preprint arXiv:1401.1408.

  19. P. M. Bleher and A. Deaño. Painlevé I double scaling limit in the cubic matrix model. Preprint arXiv:1301.4266.

  20. P. M. Bleher and A. Deaño. Topological expansion in the cubic random matrix model. Int. Math. Res. Not. IMRN, (12):2699–2755, 2013.

    MathSciNet  MATH  Google Scholar 

  21. A. Deaño, A. B. J. Kuijlaars, and P. Román. Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions. Preprint arXiv:1406.0969.

  22. A. Deaño, D. Huybrechs, and A. B. J. Kuijlaars. Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature. J. Approx. Theory, 162(12):2202–2224, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  23. P. A. Deift. Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. New York University Courant Institute of Mathematical Sciences, New York, 1999.

  24. P. Dragnev and E. B. Saff. Constrained energy problems with applications to orthogonal polynomials of a discrete variable. J. d’Analyse Mathématique, 72:229–265, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Faldey and W. Gawronski. On the limit distributions of the zeros of Jonquière polynomials and generalized classical orthogonal polynomials. Journal of Approximation Theory, 81(2):231–249, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  26. W. Gautschi. Orthogonal polynomials: Applications and computation. Acta Numerica, pages 45–119, 1996.

  27. A. A. Gončar. The convergence of Padé approximations. Mat. Sb. (N.S.), 92(134):152–164, 167, 1973.

  28. A. A. Gonchar. On convergence of Padé approximants for some classes of meromorphic functions. Math. USSR Sbornik, 26(4):555–575, 1975.

    Article  MATH  Google Scholar 

  29. A. A. Gonchar and E. A. Rakhmanov. On the convergence of simultaneous Padé approximants for systems of functions of Markov type. Trudy Mat. Inst. Steklov., 157:31–48, 234, 1981. Number theory, mathematical analysis and their applications.

  30. A. A. Gonchar and E. A. Rakhmanov. Equilibrium measure and the distribution of zeros of extremal polynomials. Mat. Sbornik, 125(2):117–127, 1984. translation from Mat. Sb., Nov. Ser. 134(176), No.3(11), 306–352 (1987).

  31. A. A. Gonchar and E. A. Rakhmanov. The equilibrium problem for vector potentials. Uspekhi Mat. Nauk, 40(4(244)):155–156, 1985.

  32. A. A. Gonchar and E. A. Rakhmanov. Equilibrium distributions and degree of rational approximation of analytic functions. Math. USSR Sbornik, 62(2):305–348, 1987. translation from Mat. Sb., Nov. Ser. 134(176), No.3(11), 306–352 (1987).

  33. A. A. Gonchar, E. A. Rakhmanov, and V. N. Sorokin. On Hermite-Padé approximants for systems of functions of Markov type. Mat. Sb., 188(5):33–58, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  34. E. Heine. Handbuch der Kugelfunctionen, volume II. G. Reimer, Berlin, \(2\)nd. edition, 1878.

  35. S. Kamvissis and E. A. Rakhmanov. Existence and regularity for an energy maximization problem in two dimensions. J. Math. Phys., 46(8):083505, 24, 2005.

  36. A. B. J. Kuijlaars and A. Martínez-Finkelshtein. Strong asymptotics for Jacobi polynomials with varying nonstandard parameters. J. Anal. Math., 94:195–234, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. B. J. Kuijlaars and G. L. F. Silva. S-curves in polynomial external fields. J. Approximation Theory, 191:1–37, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  38. G. López Lagomasino and A. Ribalta. Approximation of transfer functions of unstable infinite dimensional control systems by rational interpolants with prescribed poles. In Proceedings of the International Conference on Rational Approximation, ICRA99 (Antwerp), volume 61, pages 267–294, 2000.

  39. F. Marcellán, A. Martínez-Finkelshtein, and P. Martínez-González. Electrostatic models for zeros of polynomials: old, new, and some open problems. J. Comput. Appl. Math., 207(2):258–272, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  40. A. Martines-Finkel shteĭn, E. A. Rakhmanov, and S. P. Suetin. Differential equations for Hermite-Padé polynomials. Uspekhi Mat. Nauk, 68(1(409)):197–198, 2013. translation in Russian Math. Surveys 68 (2013), no. 1, 183–185.

  41. A. Martínez-Finkelshtein, P. Martínez-González, and R. Orive. Zeros of Jacobi polynomials with varying non-classical parameters. In Special functions (Hong Kong, 1999), pages 98–113. World Sci. Publ., River Edge, NJ, 2000.

  42. A. Martínez-Finkelshtein and R. Orive. Riemann-Hilbert analysis of Jacobi polynomials orthogonal on a single contour. J. Approx. Theory, 134(2):137–170, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  43. A. Martínez-Finkelshtein and E. A. Rakhmanov. Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials. Comm. Math. Phys., 302(1):53–111, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Martínez-Finkelshtein, P. Martínez-González, and F. Thabet. Trajectories of quadratic differentials for Jacobi polynomials with complex parameters. Preprint arXiv:1506.03434, to appear in Comput. Methods and Funct. Theory, 2015.

  45. A. Martínez-Finkelshtein and E. A. Rakhmanov. On asymptotic behavior of Heine-Stieltjes and Van Vleck polynomials. In Recent trends in orthogonal polynomials and approximation theory, volume 507 of Contemp. Math., pages 209–232. Amer. Math. Soc., Providence, RI, 2010.

  46. A. Martínez-Finkelshtein, E. A. Rakhmanov, and S. P. Suetin. Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall’s work 25 years later. In Recent advances in orthogonal polynomials, special functions, and their applications, volume 578 of Contemp. Math., pages 165–193. Amer. Math. Soc., Providence, RI, 2012.

  47. A. Martínez-Finkelshtein and G. L. F. Silva. Critical measures for vector energy: global structure of trajectories of quadratic differentials. Preprint arXiv:1509.06704.

  48. A. Máté, P. Nevai, and V. Totik. Extensions of Szegő’s theory of orthogonal polynomials. II, III. Constr. Approx., 3(1):51–72, 73–96, 1987.

  49. A. Máté, P. Nevai, and V. Totik. Strong and weak convergence of orthogonal polynomials. Amer. J. Math., 109(2):239–281, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  50. E. M. Nikishin and V. N. Sorokin. Rational approximations and orthogonality, volume 92 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Ralph P. Boas.

  51. J. Nuttall. The convergence of Padé approximants to functions with branch points. In Padé and rational approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976), pages 101–109. Academic Press, New York, 1977.

  52. J. Nuttall. Asymptotics of diagonal Hermite-Padé polynomials. J. Approx. Theory, 42(4):299–386, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  53. J. Nuttall. Asymptotics of generalized Jacobi polynomials. Constr. Approx., 2(1):59–77, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  54. J. Nuttall and S. R. Singh. Orthogonal polynomials and Padé approximants associated with a system of arcs. J. Approx. Theory, 21(1):1–42, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  55. Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark, editors. NIST handbook of mathematical functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX).

  56. O. Perron. Die Lehre von den Kettenbrüchen. Chelsea Publishing Co., New York, N. Y., 2nd. edition, 1950.

  57. E. A. Rakhmanov. Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable. Sb. Math., 187:1213–1228, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  58. E. A. Rakhmanov. On the asymptotics of Hermite-Padé polynomials for two Markov functions. Mat. Sb., 202(1):133–140, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  59. E. A. Rakhmanov. Orthogonal polynomials and \(S\)-curves. In Recent advances in orthogonal polynomials, special functions, and their applications, volume 578 of Contemp. Math., pages 195–239. Amer. Math. Soc., Providence, RI, 2012.

  60. E. A. Rakhmanov and E. A. Perevozhnikova. Variations of the equilibrium energy and \({S}\)-property of compacta of minimal capacity. Preprint, 1994.

  61. A. Ronveaux, editor. Heun’s differential equations. The Clarendon Press Oxford University Press, New York, 1995. With contributions by F. M. Arscott, S. Yu. Slavyanov, D. Schmidt, G. Wolf, P. Maroni and A. Duval.

  62. E. B. Saff and V. Totik. Logarithmic Potentials with External Fields, volume 316 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1997.

  63. E. B. Saff, J. L. Ullman, and R. S. Varga. Incomplete polynomials: an electrostatics approach. In Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980), pages 769–782. Academic Press, New York, 1980.

  64. J. F. Sánchez-Lara. Local behavior of the equilibrium measure under an external field non differentiable at a point. J. Approx. Theory, 187:1–17, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  65. B. Shapiro. Algebro-geometric aspects of Heine–Stieltjes theory. J. London Math. Soc., 83:36–56, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  66. H. Stahl. Extremal domains associated with an analytic function. I, II. Complex Variables Theory Appl., 4(4):311–324, 325–338, 1985.

  67. H. Stahl. On the divergence of certain Padé approximant and the behaviour of the associated orthogonal polynomials. In Orthogonal polynomials and applications (Bar-le-Duc, 1984), volume 1171 of Lecture Notes in Math., pages 321–330. Springer, Berlin, 1985.

  68. H. Stahl. Orthogonal polynomials with complex-valued weight function. I, II. Constr. Approx., 2(3):225–240, 241–251, 1986.

  69. H. Stahl. Asymptotics of Hermite-Padé polynomials and related convergence results—a summary of results. In Nonlinear numerical methods and rational approximation (Wilrijk, 1987), volume 43 of Math. Appl., pages 23–53. Reidel, Dordrecht, 1988.

  70. H. Stahl. The convergence of Padé approximants to functions with branch points. J. Approx. Theory, 91(2):139–204, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  71. H. Stahl. Spurious poles in Padé approximation. J. Comput. Appl. Math., 99(1–2):511–527, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  72. H. Stahl and V. Totik. General orthogonal polynomials, volume 43 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.

  73. T. J. Stieltjes. Sur certains polynômes que vérifient une équation différentielle linéaire du second ordre et sur la teorie des fonctions de Lamé. Acta Math., 6:321–326, 1885.

    Article  MathSciNet  Google Scholar 

  74. G. Szegő. Orthogonal Polynomials, volume 23 of Amer. Math. Soc. Colloq. Publ. Amer. Math. Soc., Providence, RI, fourth edition, 1975.

  75. W. Van Assche. Padé and Hermite-Padé approximation and orthogonality. Surv. Approx. Theory, 2:61–91, 2006.

    MathSciNet  MATH  Google Scholar 

  76. H. Volkmer. Multiparameter eigenvalue problems and expansion theorems. Lecture Notes Math., 1356:vi+157, 1988.

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Acknowledgments

This paper is based on a plenary talk at the FoCM conference in Montevideo, Uruguay, in 2014 by the first author (AMF), who is very grateful to the organizing committee for the invitation. This work was completed during a stay of AMF as a Visiting Chair Professor at the Department of Mathematics of the Shanghai Jiao Tong University (SJTU), China. He acknowledges the hospitality of the host department and of the SJTU. AMF was partially supported by the Spanish Government together with the European Regional Development Fund (ERDF) under Grants MTM2011-28952-C02-01 (from MICINN) and MTM2014-53963-P (from MINECO), by Junta de Andalucía (the Excellence Grant P11-FQM-7276 and the research group FQM-229), and by Campus de Excelencia Internacional del Mar (CEIMAR) of the University of Almería. We are grateful to S. P. Suetin for allowing us to use the results of his calculations in Fig. 4, as well as to an anonymous referee for several remarks that helped to improve the presentation.

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  1. Department of Mathematics, University of Almería, Almería, Spain

    A. Martínez-Finkelshtein

  2. Department of Mathematics, University of South Florida, Tampa, FL, USA

    E. A. Rakhmanov

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Martínez-Finkelshtein, A., Rakhmanov, E.A. Do Orthogonal Polynomials Dream of Symmetric Curves?. Found Comput Math 16, 1697–1736 (2016). https://doi.org/10.1007/s10208-016-9313-0

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