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Strong Asymptotics for Bergman Polynomials over Domains with Corners and Applications

Constructive Approximation Aims and scope

Abstract

Let G be a bounded simply-connected domain in the complex plane ℂ, whose boundary Γ:=∂G is a Jordan curve, and let \(\{p_{n}\}_{n=0}^{\infty}\) denote the sequence of Bergman polynomials of G. This is defined as the unique sequence

$$p_n(z) = \lambda_n z^n+\cdots, \quad \lambda_n>0,\ n=0,1,2,\ldots, $$

of polynomials that are orthonormal with respect to the inner product

$$\langle f,g\rangle := \int_G f(z) \overline{g(z)} \,dA(z), $$

where dA stands for the area measure.

We establish the strong asymptotics for p n and λ n , n∈ℕ, under the assumption that Γ is piecewise analytic. This complements an investigation started in 1923 by T. Carleman, who derived the strong asymptotics for Γ analytic, and carried over by P.K. Suetin in the 1960s, who established them for smooth Γ. In order to do so, we use a new approach based on tools from quasiconformal mapping theory. The impact of the resulting theory is demonstrated in a number of applications, varying from coefficient estimates in the well-known class Σ of univalent functions and a connection with operator theory, to the computation of capacities and a reconstruction algorithm from moments.

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References

  1. Ahlfors, L.V.: Quasiconformal reflections. Acta Math. 109, 291–301 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahlfors, L.V.: Lectures on Quasiconformal Mappings. Van Nostrand Mathematical Studies, vol. 10. D. Van Nostrand Co., Inc., Toronto (1966). Manuscript prepared with the assistance of Clifford J. Earle Jr.

    MATH  Google Scholar 

  3. Andrievskii, V.V., Blatt, H.-P.: Discrepancy of Signed Measures and Polynomial Approximation. Springer Monographs in Mathematics. Springer, New York (2002)

    Book  MATH  Google Scholar 

  4. Andrievskii, V.V., Belyi, V.I., Dzjadyk, V.K.: Conformal Invariants in Constructive Theory of Functions of Complex Variable. Advanced Series in Mathematical Science and Engineering, vol. 1. World Federation Publishers Company, Atlanta (1995)

    MATH  Google Scholar 

  5. Arnoldi, W.E.: The principle of minimized iteration in the solution of the matrix eigenvalue problem. Q. Appl. Math. 9, 17–29 (1951)

    MathSciNet  MATH  Google Scholar 

  6. Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton Mathematical Series, vol. 48. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  7. Axler, S., Conway, J.B., McDonald, G.: Toeplitz operators on Bergman spaces. Can. J. Math. 34(2), 466–483 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Baratchart, L., Martínez-Finkelshtein, A., Jimenez, D., Lubinsky, D.S., Mhaskar, H.N., Pritsker, I., Putinar, M., Stylianopoulos, N., Totik, V., Varju, P., Xu, Y.: Open problems in constructive function theory. Electron. Trans. Numer. Anal. 25, 511–525 (2006) (electronic)

    MathSciNet  MATH  Google Scholar 

  9. Belyĭ, V.I.: Conformal mappings and approximation of analytic functions in domains with quasiconformal boundary. Math. USSR Sb. 31, 289–317 (1977)

    Article  MATH  Google Scholar 

  10. Böttcher, A., Grudsky, S.M.: Spectral Properties of Banded Toeplitz Matrices. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2005)

    Book  MATH  Google Scholar 

  11. Carleman, T.: Über die Approximation analytisher Funktionen durch lineare Aggregate von vorgegebenen Potenzen. Ark. Mat. Astron. Fys. 17(9), 215–244 (1923)

    Google Scholar 

  12. Clunie, J.: On schlicht functions. Ann. Math. 69, 511–519 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  13. Davis, P., Pollak, H.: On the analytic continuation of mapping functions. Trans. Am. Math. Soc. 87, 198–225 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137(2), 295–368 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Duren, P.L.: Theory of H p Spaces. Pure and Applied Mathematics, vol. 38. Academic Press, New York (1970)

    MATH  Google Scholar 

  16. Duren, P.L.: Univalent Functions. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259. Springer, New York (1983)

    MATH  Google Scholar 

  17. Fabijonas, B.R., Olver, F.W.J.: On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM Rev. 41(4), 762–773 (1999) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fokas, A.S., Its, A.R., Kitaev, A.V.: Discrete Painlevé equations and their appearance in quantum gravity. Commun. Math. Phys. 142(2), 313–344 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fokas, A.S., Its, A.R., Kitaev, A.V.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147(2), 395–430 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gaier, D.: Lectures on Complex Approximation. Birkhäuser, Boston (1987)

    Book  MATH  Google Scholar 

  21. Gaier, D.: The Faber operator and its boundedness. J. Approx. Theory 101(2), 265–277 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gaier, D.: On the decrease of Faber polynomials in domains with piecewise analytic boundary. Analysis (Munich) 21(2), 219–229 (2001)

    MathSciNet  MATH  Google Scholar 

  23. Gragg, W.B., Reichel, L.: On the application of orthogonal polynomials to the iterative solution of linear systems of equations with indefinite or non-Hermitian matrices. Linear Algebra Appl. 88/89, 349–371 (1987)

    Article  MathSciNet  Google Scholar 

  24. Gustafsson, B., He, C., Milanfar, P., Putinar, M.: Reconstructing planar domains from their moments. Inverse Probl. 16(4), 1053–1070 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gustafsson, B., Putinar, M., Saff, E., Stylianopoulos, N.: Bergman polynomials on an archipelago: estimates, zeros and shape reconstruction. Adv. Math. 222, 1405–1460 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Henrici, P.: Applied and Computational Complex Analysis. Pure and Applied Mathematics (New York), vol. 3. Wiley, New York (1986)

    MATH  Google Scholar 

  27. Johnston, E.R.: A study in polynomial approximation in the complex domain. Ph.D. thesis, University of Minnesota, March (1954)

  28. Khavinson, D.: Remarks concerning boundary properties of analytic functions of E p -classes. Indiana Univ. Math. J. 31(6), 779–787 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  29. Khavinson, D., Lundberg, E.: The search for singularities of solutions to the Dirichlet problem: recent developments. CRM Proc. Lect. Notes 51, 121–132 (2010)

    MathSciNet  Google Scholar 

  30. Khavinson, D., Stylianopoulos, N.: Recurrence relations for orthogonal polynomials and algebraicity of solutions of the Dirichlet problem, around the research of Vladimir Maz’ya II. In: Int. Math. Ser. (N. Y.), vol. 12, pp. 219–228. Springer, New York (2010)

    Google Scholar 

  31. Lehman, R.S.: Development of the mapping function at an analytic corner. Pac. J. Math. 7, 1437–1449 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane, 2nd edn. Springer, New York (1973)

    Book  MATH  Google Scholar 

  33. Lubinsky, D.S.: A new approach to universality limits involving orthogonal polynomials. Ann. Math. 170(2), 915–939 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lytrides, M., Stylianopoulos, N.S.: Error analysis of the Bergman kernel method with singular basis functions. Comput. Methods Funct. Theory 11(2), 487–526 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Maymeskul, V.V., Saff, E.B., Stylianopoulos, N.S.: L 2-approximations of power and logarithmic functions with applications to numerical conformal mapping. Numer. Math. 91(3), 503–542 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Miña-Díaz, E.: An asymptotic integral representation for Carleman orthogonal polynomials. Int. Math. Res. Not. IMRN 2008(16), rnn065 (2008). 38

    Google Scholar 

  37. Papamichael, N., Warby, M.K.: Stability and convergence properties of Bergman kernel methods for numerical conformal mapping. Numer. Math. 48(6), 639–669 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pommerenke, Ch.: Univalent Functions. Vandenhoeck & Ruprecht, Göttingen (1975). With a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. XXV.

    MATH  Google Scholar 

  39. Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Fundamental Principles of Mathematical Sciences, vol. 299. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  40. Putinar, M., Stylianopoulos, N.: Finite-term relations for planar orthogonal polynomials. Complex Anal. Oper. Theory 1(3), 447–456 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rosenbloom, P.C., Warschawski, S.E.: Approximation by Polynomials, Lectures on Functions of a Complex Variable, pp. 287–302. The University of Michigan Press, Ann Arbor (1955)

    Google Scholar 

  42. Saff, E.B.: Orthogonal polynomials from a complex perspective. In: Orthogonal Polynomials (Columbus, OH, 1989), pp. 363–393. Kluwer Acad. Publ., Dordrecht (1990)

    Chapter  Google Scholar 

  43. Saff, E.B., Stylianopoulos, N.S.: Asymptotics for Hessenberg matrices for the Bergman shift operator on Jordan regions. Complex Anal. Oper. Theory (in press). doi:10.1007/s1178-012-0252-8

  44. Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Springer, Berlin (1997)

    MATH  Google Scholar 

  45. Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1. American Mathematical Society Colloquium Publications, vol. 54. American Mathematical Society, Providence (2005). Classical theory

    MATH  Google Scholar 

  46. Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 2. American Mathematical Society Colloquium Publications, vol. 54. American Mathematical Society, Providence (2005). Spectral theory

    MATH  Google Scholar 

  47. Stahl, H., Totik, V.: General Orthogonal Polynomials. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  48. Stylianopoulos, N.: Strong asymptotics for Bergman polynomials over non-smooth domains. C. R. Math. Acad. Sci. Paris 348(1–2), 21–24 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Suetin, P.K.: Fundamental properties of polynomials orthogonal on a contour. Usp. Mat. Nauk 21(2(128)), 41–88 (1966)

    MathSciNet  Google Scholar 

  50. Suetin, P.K.: Polynomials Orthogonal over a Region and Bieberbach Polynomials. American Mathematical Society, Providence (1974)

    Google Scholar 

  51. Szegő, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society Colloquium Publications, vol. XXIII. American Mathematical Society, Providence (1975)

    Google Scholar 

  52. Taylor, J.M.: The condition of gram matrices and related problems. Proc. R. Soc. Edinb. A 80(1–2), 45–56 (1978)

    Article  MATH  Google Scholar 

  53. Widom, H.: Polynomials associated with measures in the complex plane. J. Math. Mech. 16, 997–1013 (1967)

    MathSciNet  MATH  Google Scholar 

  54. Widom, H.: Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3, 127–232 (1969)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Nikos Stylianopoulos.

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Communicated by Edward B. Saff.

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Stylianopoulos, N. Strong Asymptotics for Bergman Polynomials over Domains with Corners and Applications. Constr Approx 38, 59–100 (2013). https://doi.org/10.1007/s00365-012-9174-y

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