Journal of Mathematical Sciences

, Volume 218, Issue 5, pp 678–698 | Cite as

On the Best Linear Approximation of Holomorphic Functions

  • Yu. A. FarkovEmail author


Let Ω be an open subset of the complex plane and let E be a compact subset of Ω. The present survey is concerned with linear n-widths for the class H (Ω) in the space C(E) and some problems on the best linear approximation of classes of Hardy–Sobolev-type in L p -spaces. It is known that the partial sums of the Faber series give the classical method for approximation of functions f ∈ H (Ω) in the metric of C(E) when E is a bounded continuum with simply connected complement and Ω is a canonical neighborhood of E. Generalizations of the Faber series are defined for the case where Ω is a multiply connected domain or a disjoint union of several such domains, while E can be split into a finite number of continua. The exact values of n-widths and asymptotic formulas for the ε-entropy of classes of holomorphic functions with bounded fractional derivatives in domains of tube type are presented. Also, some results about Faber’s approximations in connection with their applications in numerical analysis are mentioned.


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  1. 1.
    V. Andrievskii and H.-P. Blatt, “On the distribution of zeros of Faber polynomials,” Comput. Methods Funct. Theory, 11, 263–282 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    K. I. Babenko, Foundations of Numerical Analysis [in Russian], Nauka, Moscow (1986).Google Scholar
  3. 3.
    V. I. Belyi, “Modern methods of the geometric theory of functions of a complex variable in approximation problems,” Algebra Analiz, 9, No. 3, 3–40 (1997).Google Scholar
  4. 4.
    V. N. Belykh, “Estimates of Kolmogorov’s ε-entropy for compact sets of infinitely differentiable aperiodic functions (Babenko’s problem),” Dokl. Math., 88, No. 2, 503–507 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. G. Borisov and S. V. Shabanov, “Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media,” J. Comput. Phys., 216, 391–402 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Bouali, “Faber polynomials, Cayley–Hamilton equation and Newton symmetric functions,” Bull. Sci. Math., 130, 49–70 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    I. Bruj and G. Schmieder, “Best approximation and saturation on domains bounded by curves of bounded rotation,” J. Approx. Theory, 100, 157–182 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Carney, A. Etropolski, and S. Pitman, “Powers of the eta-function and Hecke operators,” Int. J. Number Theory, 8, 599–611 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    O. Devys, “Faber polynomials and spectrum localisation,” Comput. Methods Funct. Theory, 13, 107–131 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    H. Ding, K. I. Gross, and D. S. P. Richards, “The N-widths of spaces of holomorphic functions on bounded symmetric domains of tube type. I,” J. Approx. Theory, 104, 121–141 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    L. K. Dodunova and S. A. Savikhin, “Completeness of subsystems of Faber polynomials,” Izv. Vyssh. Uchebn. Zaved. Mat., 9, 3–7 (2012).MathSciNetzbMATHGoogle Scholar
  12. 12.
    M. M. Dragilev, “On common bases of the spaces A(G) and \( \overline{A}\left(\overline{G}\right) \),” Sib. Mat. Zh., 40, No. 1, 69–74 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    P. L. Duren, “Distortion in certain conformal mappings of an annulus,” Michigan Math. J., 10, No. 4, 431–441 (1963).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    E. M. Dyn’kin, “The rate of polynomial approximation in the complex domain,” in: Complex Analysis and Spectral Theory, Lect. Notes Math., Vol. 864, Springer, Berlin (1981), pp. 90–142.Google Scholar
  15. 15.
    E. M. Dyn’kin, “A constructive characterization of classes of S. L. Sobolev and O. V. Besov,” Proc. Steklov Inst. Math. 155, 39–74 (1983).zbMATHGoogle Scholar
  16. 16.
    V. D. Erokhin, “On asymptotics of the ε-entropy of analytic functions,” Dokl. Akad. Nauk SSSR, 120, No. 5, 949–952 (1958).MathSciNetGoogle Scholar
  17. 17.
    V. D. Erokhin, “Some theorems on analytic mapping of multiconnected domains,” Usp. Mat. Nauk, 15, No. 4, 203–204 (1960).Google Scholar
  18. 18.
    V. D. Erokhin, “Estimates of the ε-entropy and linear widths of some classes of analytic functions,” in: Markushevich A. I., ed., Researches in Modern Problems of Function Theory of a Complex Variable Fizmatgiz, Moscow (1961), pp. 159–167.Google Scholar
  19. 19.
    V. D. Erokhin, “Best linear approximations of functions analytically continuable from a given continuum into a given region,” Usp. Mat. Nauk, 23, No. 1, 91–132 (1968).Google Scholar
  20. 20.
    J. Faraut and A. Koranyi, Analysis on Symmetric Cones, Oxford Univ. Press, New York (1994).Google Scholar
  21. 21.
    Yu. A. Farkov, “Asymptotic properties of generalized Faber–Erokhin basis functions,” Sib. Mat. Zh., 22, No. 4, 173–189 (1981).Google Scholar
  22. 22.
    Yu. A. Farkov, “Faber–Erokhin basis functions of several variables and estimates of ε-entropy,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 81–88 (1982).Google Scholar
  23. 23.
    Yu. A. Farkov, “Faber–Erokhin operators and isomorphisms of some spaces of analytic functions,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 7, 81–83 (1982).Google Scholar
  24. 24.
    Yu. A. Farkov, “Faber–Erokhin basic functions in the neighborhood of several continua,” Math. Notes, 36, No. 6, 883–892 (1984).Google Scholar
  25. 25.
    Yu. A. Farkov, “Widths of classes of analytic functions with bounded derivatives,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 4, 84–86 (1988).Google Scholar
  26. 26.
    Yu. A. Farkov, “Widths of Hardy classes and Bergman classes on the ball in n,” Usp. Mat. Nauk, 45, No. 5, 197–198 (1990).MathSciNetzbMATHGoogle Scholar
  27. 27.
    Yu. A. Farkov, “The N-widths of Hardy–Sobolev spaces of several complex variables,” J. Approx. Theory, 75, 183–197 (1993).Google Scholar
  28. 28.
    Yu. A. Farkov, “n-widths, Faber expansion, and computation of analytic functions,” J. Complexity, 12, 58–79 (1996).Google Scholar
  29. 29.
    Yu. A. Farkov, “On the ε-entropy of classes of holomorphic functions,” Math. Notes, 68, No. 2, 286–293 (2000).Google Scholar
  30. 30.
    N. A. Farkova, “The use of Faber polynomials to solve systems of linear equations,” Zh. Vychisl. Mat. Mat. Fiz., 28, No. 11, 1634–1648 (1988).MathSciNetzbMATHGoogle Scholar
  31. 31.
    N. A. Farkova, “Application of Faber polynomials to the calculation of eigenvalues,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 9, 65–72 (1992).Google Scholar
  32. 32.
    S. D. Fisher and C. A. Micchelli, “The n-width of sets of analytic functions,” Duke Math. J., 47, 789–801 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    L. Frerick and J. M¨uller, “Polynomial approximation on compact sets bounded by Dini-smooth arcs,” Comput. Methods Funct. Theory, 3, 273–284 (2003).Google Scholar
  34. 34.
    D. Gaier, “On the decrease of Faber polynomials in domains with piecewise analytic boundary,” Analysis, 21, 219–229 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    T. H. Ganelius, “Rational approximation in the complex plane and on the line,” Ann. Acad. Sci. Fenn. Ser. A. I. Math., 2, 129–145 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    I. Garaus, “The numerical solution for system of singular integro-differential equations by Faber–Laurent polynomials,” in: Zh. Li, L. Vulkov, and J. Wásniewski, eds., Numerical Analysis and Its Applications: Third Int. Conf., NAA 2004, Rousse, Bulgaria, June 29–July 3, 2004, Revised Selected Papers, Lect. Notes Comput. Sci., Vol. 3401, Springer, Berlin (2005), pp. 219–223.Google Scholar
  37. 37.
    P. Henrici, Applied and Computational Complex Analysis, Wiley, New York (1986).zbMATHGoogle Scholar
  38. 38.
    O. Hübner, “Die Faktorisierung konformer Abbildungen und Anwendungen,” Math. Z., 99, 193–206 (1967).MathSciNetCrossRefGoogle Scholar
  39. 39.
    S. Z. Jafarov, “The inverse theorem of approximation theory in Smirnov–Orlicz classes,” Math. Inequal. Appl., 15, No. 4, 835–844 (2012).MathSciNetzbMATHGoogle Scholar
  40. 40.
    V. N. Konovalov, “To the problem of widths of classes of analytic functions,” Ukr. Mat. Zh., 30, No. 5, 668–670 (1978).MathSciNetzbMATHGoogle Scholar
  41. 41.
    P. Novati, “Solving linear initial value problems by Faber polynomials,” Numer. Linear Algebra Appl., 10, 247–270 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    K. Yu. Osipenko, “Optimal interpolation of analytic functions,” Math. Notes, 12, No. 4, 712–719 (1972).Google Scholar
  43. 43.
    K. Yu. Osipenko, “Exact n-widths of Hardy–Sobolev classes,” Constr. Approx., 13, No. 1, 17–27 (1997).Google Scholar
  44. 44.
    A. A. Pekarskii, “Rational approximations of functions with derivative in a V. I. Smirnov space,” Algebra Analiz, 13, No. 2, 165–190 (2001).MathSciNetGoogle Scholar
  45. 45.
    V. V. Peller, “Rational approximation in L p and Faber transforms,” Zap. Nauchn. Sem. LOMI, 157, 70–75 (1987).MathSciNetGoogle Scholar
  46. 46.
    W. Rudin, Function theory in the unit ball of ℂ n, Springer, New York (1980).Google Scholar
  47. 47.
    S. V. Shvedenko, “Hardy classes and related spaces of analytic functions in the unit disc, polydisc and ball,” J. Sov. Math., 39, No. 6, 3011–3087 (1987).CrossRefzbMATHGoogle Scholar
  48. 48.
    V. I. Smirnov and N. A. Lebedev, Functions of a Complex Variable. Constructive Theory, MIT Press, Cambridge, MA (1968).Google Scholar
  49. 49.
    G. Starke and R. S. Varga, “A hybrid Arnoldi–Faber iterative method for nonsymmetric systems of linear equations,” Numer. Math., 64, 231–240 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    P. K. Suetin, “Order comparison of various norms of polynomials in a complex domain,” Mat. Zap. Ural’sk. Univ., 5, No. 4, 91–100 (1966).MathSciNetGoogle Scholar
  51. 51.
    P. K. Suetin, Series of Faber Polynomials [in Russian], Nauka, Moscow (1984); Gordon and Breach (1998).Google Scholar
  52. 52.
    S. P. Suetin, “Padé approximants and efficient analytic continuation of a power series,” Russ. Math. Surv., 57, No. 1, 43–141 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    V. M. Tikhomirov, “Approximation theory,” in: R. V. Gamkrelidze, ed., Analysis-II. Convex Analysis and Approximation Theory, Encycl. Math. Sci., Vol. 14, Springer, Berlin (1990), pp. 93–243.Google Scholar
  54. 54.
    V. Totik, “Chebyshev polynomials on a system of curves,” J. Anal. Math., 118, No. 1, 317–338 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    M. M. Tsvil’, “On the convergence of ball means of double Fourier–Faber series,” Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Reg. Estestv. Nauki., 11, 63–69 (2005).Google Scholar
  56. 56.
    S. B. Vakarchuk and M. Sh. Shabozov, “The widths of classes of analytic functions in a disc,” Mat. Sb., 201, No. 8, 3–22 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    J. L. Walsh, “Sur l’approximation par fonctions rationnelles et par fonctions holomorphes bornées,” Ann. Mat., 39, No. 4, 267–277 (1955).MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    J. L. Walsh, “On the conformal mapping of multiply connected regions,” Trans. Am. Math. Soc., 82, No. 1, 128–146 (1956).MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    J. L. Walsh, “A generalization of Faber’s polynomials,” Math. Ann., 136, No. 1, 23–33 (1958).MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc., Providence (1960).Google Scholar
  61. 61.
    J. L. Walsh and W. E. Sewell, “Sufficient conditions for various degrees of approximation by polynomials,” Duke Math. J., 6, No. 3, 658–705 (1940).MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    P. Wójcik, M. A. Sheshko, and S. M. Sheshko, “Application of Faber polynomials to the approximate solution of singular integral equations with the Cauchy kernel,” Differ. Equ., 49, No. 2, 198–209 (2013).Google Scholar
  63. 63.
    V. P. Zaharyuta, “Kolmogorov problem on widths asymptotics and pluripotential theory,” in: A. Aytuna, ed., Functional Analysis and Complex Analysis: Functional Analysis and Complex Analysis, September 17–21, 2007, Sabancı University, ˙Istanbul, Turkey, Contemp. Math., Vol. 481, Amer. Math. Soc. (2009), pp. 171–196.Google Scholar
  64. 64.
    V. Zaharyuta, “On asymptotics of entropy of a class of analytic functions,” Funct. Approx. Comment. Math., 44, No. 2, 307–315 (2011).MathSciNetCrossRefGoogle Scholar
  65. 65.
    J. Zhang, “Symbolic and numerical computation on Bessel functions of complex argument and large magnitude,” J. Comput. Appl. Math. 75, 99–118 (1996).MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Russian State Geological Prospecting UniversityMoscowRussia
  2. 2.Russian Presidential Academy of National Economy and Public AdministrationMoscowRussia

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