Method of interior variations and existence of S-compact sets

  • V. I. Buslaev
  • A. Martínez-Finkelshtein
  • S. P. Suetin
Article

Abstract

The variation of equilibrium energy is analyzed for three different functionals that naturally arise in solving a number of problems in the theory of constructive rational approximation of multivalued analytic functions. The variational approach is based on the relationship between the variation of the equilibrium energy and the equilibrium measure. In all three cases the following result is obtained: for the energy functional and the class of admissible compact sets corresponding to the problem, the arising stationary compact set is fully characterized by a certain symmetry property.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. I. Aptekarev, “Asymptotics of Hermite-Padé Approximants for Two Functions with Branch Points,” Dokl. Akad. Nauk 422(4), 443–445 (2008) [Dokl. Math. 78 (2), 717–719 (2008)].Google Scholar
  2. 2.
    A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, and S. P. Suetin, “Padé Approximants, Continued Fractions, and Orthogonal Polynomials,” Usp. Mat. Nauk 66(6), 37–122 (2011) [Russ. Math. Surv. 66, 1049–1131 (2011)].Google Scholar
  3. 3.
    A. I. Aptekarev and R. Khabibullin, “Asymptotic Expansions for Polynomials Orthogonal with Respect to a Complex Non-constant Weight Function,” Tr. Mosk. Mat. Obshch. 68, 3–43 (2007) [Trans. Moscow Math. Soc. 2007, 1–37 (2007)].MathSciNetGoogle Scholar
  4. 4.
    A. I. Aptekarev, A. B. J. Kuijlaars, and W. Van Assche, “Asymptotics of Hermite-Padé Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0),” Int. Math. Res. Pap., doi: 10.1093/imrp/rpm007 (2008).Google Scholar
  5. 5.
    A. I. Aptekarev, V. G. Lysov, and D. N. Tulyakov, “Random Matrices with External Source and the Asymptotic Behaviour of Multiple Orthogonal Polynomials,” Mat. Sb. 202(2), 3–56 (2011) [Sb. Math. 202, 155–206 (2011)].MathSciNetCrossRefGoogle Scholar
  6. 6.
    T. Bloom, “Weighted Approximation in CN,” in E. B. Saff and V. Totik, Logarithmic Potentials with External Fields (Springer, Berlin, 1997), Appendix B, Grundl. Math. Wiss. 316, pp. 465–481.Google Scholar
  7. 7.
    V. I. Buslaev, “On the Convergence of Continued T-Fractions,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 235, 36–51 (2001) [Proc. Steklov Inst. Math. 235, 29–43 (2001)].MathSciNetGoogle Scholar
  8. 8.
    V. I. Buslaev, “The Convergence of Multipoint Padé Approximants for Piecewise Analytic Functions,” Mat. Sb. 204(2) (2013) [Sb. Math. 204 (2) (2013)].Google Scholar
  9. 9.
    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).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    V. N. Dubinin, “Symmetrization in the Geometric Theory of Functions of a Complex Variable,” Usp. Mat. Nauk 49(1), 3–76 (1994) [Russ. Math. Surv. 49 (1), 1–79 (1994)].MathSciNetGoogle Scholar
  11. 11.
    V. N. Dubinin, “Some Properties of the Reduced Inner Modulus,” Sib. Mat. Zh. 35(4), 774–792 (1994) [Sib. Math. J. 35, 689–705 (1994)].MathSciNetCrossRefGoogle Scholar
  12. 12.
    V. N. Dubinin, Capacities of Condensers and Symmetrization in Geometric Theory of Functions of a Complex Variable (Dal’nauka, Vladivostok, 2009) [in Russian].Google Scholar
  13. 13.
    V. N. Dubinin, D. B. Karp, and V. A. Shlyk, “Selected Problems of Geometric Function Theory and Potential Theory,” Dal’nevost. Mat. Zh. 8(1), 46–95 (2008).MathSciNetGoogle Scholar
  14. 14.
    A. A. Gonchar, “Rational Approximation of Analytic Functions,” Sovrem. Probl. Mat. 1, 83–106 (2003) [Proc. Steklov Inst. Math. 272 (Suppl. 2), S44–S57 (2011)].MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. A. Gonchar and E. A. Rakhmanov, “On Convergence of Simultaneous Padé Approximants for Systems of Functions of Markov Type,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 157, 31–48 (1981) [Proc. Steklov Inst. Math. 157, 31–50 (1983)].MathSciNetMATHGoogle Scholar
  16. 16.
    A. A. Gonchar and E. A. Rakhmanov, “Equilibrium Measure and the Distribution of Zeros of Extremal Polynomials,” Mat. Sb. 125(1), 117–127 (1984) [Math. USSR, Sb. 53, 119–130 (1986)].MathSciNetGoogle Scholar
  17. 17.
    A. A. Gonchar and E. A. Rakhmanov, “On the Equilibrium Problem for Vector Potentials,” Usp. Mat. Nauk 40(4), 155–156 (1985) [Russ. Math. Surv. 40 (4), 183–184 (1985)].MathSciNetGoogle Scholar
  18. 18.
    A. A. Gonchar and E. A. Rakhmanov, “Equilibrium Distributions and Degree of Rational Approximation of Analytic Functions,” Mat. Sb. 134(3), 306–352 (1987) [Math. USSR, Sb. 62, 305–348 (1989)].Google Scholar
  19. 19.
    A. A. Gonchar, E. A. Rakhmanov, and S. P. Suetin, “On the Convergence of Padé Approximations of Orthogonal Expansions,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 200, 136–146 (1991) [Proc. Steklov Inst. Math. 200, 149–159 (1993)].Google Scholar
  20. 20.
    A. A. Gonchar, E. A. Rakhmanov, and S. P. Suetin, “On the Rate of Convergence of Padé Approximants of Orthogonal Expansions,” in Progress in Approximation Theory: An International Perspective (Springer, New York, 1992), pp. 169–190.CrossRefGoogle Scholar
  21. 21.
    A. A. Gonchar, E. A. Rakhmanov, and S. P. Suetin, “On the Convergence of Chebyshev-Padé Approximations to Real-Valued Algebraic Functions,” arXiv: 1009.4813 [math.CV].Google Scholar
  22. 22.
    A. A. Gonchar, E. A. Rakhmanov, and S. P. Suetin, “Padé-Chebyshev Approximants of Multivalued Analytic Functions, Variation of Equilibrium Energy, and the S-Property of Stationary Compact Sets,” Usp. Mat. Nauk 66(6), 3–36 (2011) [Russ. Math. Surv. 66, 1015–1048 (2011)].MathSciNetGoogle Scholar
  23. 23.
    S. Kamvissis and E. A. Rakhmanov, “Existence and Regularity for an Energy Maximization Problem in Two Dimensions,” J. Math. Phys. 46(8), 083505 (2005).MathSciNetCrossRefGoogle Scholar
  24. 24.
    A. B. J. Kuijlaars and A. Martínez-Finkelshtein, “Strong Asymptotics for Jacobi Polynomials with Varying Nonstandard Parameters,” J. Anal. Math. 94, 195–234 (2004).MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    G. V. Kuz’mina, Moduli of Families of Curves and Quadratic Differentials (Nauka, Leningrad, 1980), Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 139 [Proc. Steklov Inst. Math. 139 (1982)].Google Scholar
  26. 26.
    G. V. Kuz’mina, “Methods of Geometric Function Theory. I,” Algebra Anal. 9(3), 41–103 (1997) [St. Petersburg Math. J. 9, 455–507 (1998)].MathSciNetGoogle Scholar
  27. 27.
    G. V. Kuz’mina, “Methods of Geometric Function Theory. II,” Algebra Anal. 9(5), 1–50 (1997) [St. Petersburg Math. J. 9, 889–930 (1998)].MathSciNetGoogle Scholar
  28. 28.
    G. V. Kuz’mina, “Gennadii Mikhailovich Goluzin and Geometric Function Theory,” Algebra Anal. 18(3), 3–38 (2006) [St. Petersburg Math. J. 18, 347–372 (2007)].MathSciNetGoogle Scholar
  29. 29.
    N. S. Landkof, Foundations of Modern Potential Theory (Nauka, Moscow, 1966; Springer, Berlin, 1972).Google Scholar
  30. 30.
    M. A. Lapik, “Support of the Extremal Measure in a Vector Equilibrium Problem,” Mat. Sb. 197(8), 101–118 (2006) [Sb. Math. 197, 1205–1221 (2006)].MathSciNetCrossRefGoogle Scholar
  31. 31.
    A. Martínez-Finkelshtein, “Trajectories of Quadratic Differentials and Approximations of Exponents on the Semiaxis,” in Complex Methods in Approximation Theory: Proc. Workshop, Almería (Spain), 1995 (Univ. Almería, Almería, 1997), Monogr. Cienc. Tecnol. 2, pp. 69–84.Google Scholar
  32. 32.
    A. Martínez-Finkelshtein, P. Martínez-González, and R. Orive, “Zeros of Jacobi Polynomials with Varying Nonclassical Parameters,” in Special Functions: Proc. Workshop, Hong Kong, 1999 (World Sci., Singapore, 2000), pp. 98–113.Google Scholar
  33. 33.
    A. Martínez-Finkelshtein, P. Martínez-González, and R. Orive, “On Asymptotic Zero Distribution of Laguerre and Generalized Bessel Polynomials with Varying Parameters,” J. Comput. Appl. Math. 133(1–2), 477–487 (2001).MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    A. Martínez-Finkelshtein and R. Orive, “Riemann-Hilbert Analysis for Jacobi Polynomials Orthogonal on a Single Contour,” J. Approx. Theory 134(2), 137–170 (2005).MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    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 (Am. Math. Soc., Providence, RI, 2010), Contemp. Math. 507, pp. 209–232.Google Scholar
  36. 36.
    A. Martínez-Finkelshtein and E. A. Rakhmanov, “Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials,” Commun. Math. Phys. 302(1), 53–111 (2011); arXiv: 0902.0193 [math.CA].MATHCrossRefGoogle Scholar
  37. 37.
    A. Martínez-Finkelshtein, E. A. Rakhmanov, and S. P. Suetin, “Variation of the Equilibrium Measure and the S-Property of a Stationary Compact Set,” Usp. Mat. Nauk 66(1), 183–184 (2011) [Russ. Math. Surv. 66, 176–178 (2011)].Google Scholar
  38. 38.
    A. Martínez-Finkelshtein, E. A. Rakhmanov, and S. P. Suetin, “Variation of the Equilibrium Energy and the S-Property of Stationary Compact Sets,” Mat. Sb. 202(12), 113–136 (2011) [Sb. Math. 202, 1831–1852 (2011)].CrossRefGoogle Scholar
  39. 39.
    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 (Am. Math. Soc., Providence, RI, 2012), Contemp. Math. 578, pp. 165–193.CrossRefGoogle Scholar
  40. 40.
    H. N. Mhaskar and E. B. Saff, “Extremal Problems for Polynomials with Exponential Weights,” Trans. Am. Math. Soc. 285(1), 203–234 (1984).MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    H. N. Mhaskar and E. B. Saff, “Weighted Polynomials on Finite and Infinite Intervals: A Unified Approach,” Bull. Am. Math. Soc. 11(2), 351–354 (1984).MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    E. A. Perevoznikova and E. A. Rakhmanov, “Variation of the Equilibrium Energy and the S-Property of Compact Sets of Minimum Capacity,” Preprint (Moscow, 1994).Google Scholar
  43. 43.
    E. A. Rakhmanov, “Orthogonal Polynomials and S-Curves,” in Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications (Am. Math. Soc., Providence, RI, 2012), Contemp. Math. 578, pp. 195–239; arXiv: 1112.5713 [math.CV].Google Scholar
  44. 44.
    H. Stahl, “Extremal Domains Associated with an Analytic Function. I,” Complex Variables, Theory Appl. 4, 311–324 (1985).MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    H. Stahl, “Extremal Domains Associated with an Analytic Function. II,” Complex Variables, Theory Appl. 4, 325–338 (1985).MATHCrossRefGoogle Scholar
  46. 46.
    H. Stahl, “The Structure of Extremal Domains Associated with an Analytic Function,” Complex Variables, Theory Appl. 4, 339–354 (1985).MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    H. Stahl, “Orthogonal Polynomials with Complex-Valued Weight Function. I,” Constr. Approx. 2, 225–240 (1986).MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    H. Stahl, “Orthogonal Polynomials with Complex-Valued Weight Function. II,” Constr. Approx. 2, 241–251 (1986).MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    H. Stahl, “The Convergence of Padé Approximants to Functions with Branch Points,” J. Approx. Theory 91(2), 139–204 (1997).MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    S. P. Suetin, “An Analogue of the Hadamard and Schiffer Variational Formulas,” Teor. Mat. Fiz. 170(3), 335–341 (2012) [Theor. Math. Phys. 170, 274–279 (2012)].CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • V. I. Buslaev
    • 1
  • A. Martínez-Finkelshtein
    • 2
  • S. P. Suetin
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Universidad de AlmeríaAlmeríaSpain

Personalised recommendations