Overconvergence in ℂ of Some Bernstein-Type Operators

  • Sorin G. Gal


Section 1.1 of this chapter contains classical definitions and results in complex analysis useful for the next sections.


  1. 1.
    Abel, U., Gupta, V., Mohapatra, R.N.: Local approximation by beta operators. Nonlinear Anal. 62(1), 41–52 (2005)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Abel, U., Gupta, V., Mohapatra, R.: Local approximation by a variant of Bernstein-Durrmeyer operators. Nonlinear Anal. Theory Methods Appl. 68(11), 3372–3381 (2008)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Abel, U., Heilmann, H.: The complete asymptotic expansion for Bernstein-Durrmeyer operator with Jacobi weights. Mediterr. J. Math. 1(4), 487–499 (2004)MathSciNetMATHCrossRefGoogle Scholar
  4. 5.
    Alexander, J.W.: Functions which map the interior of the unit circle upon simple regions. Ann. Math. 17(1915), 12–22 (1915)MATHCrossRefGoogle Scholar
  5. 7.
    Altomare, F., Mangino, E.: On a generalization of Baskakov operator. Rev. Roumaine Math. Pures Appl. 44(5–6), 683–705 (1999)MathSciNetMATHGoogle Scholar
  6. 8.
    Altomare, F., Raşa, I.: Feller semigroups, Bernstein-type operators and generalized convexity associated with positive projections. New Develpments in Aproximation Theory, Dortmund, 1998, pp. 9–32. Birkhauser, Basel 1999 (1999)Google Scholar
  7. 9.
    Anastassiou, G., Gal, S.G.: Approximation by complex Bernstein-Schurer and Kantorovich-Schurer polynomials in compact disks. Comput. Math. Appl. 58(4), 734–743 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 11.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  9. 12.
    Bartolomeu, J., He, M.: On Faber polynomials generated by an m-star. Math. Comput. 62, 277–288 (1994)Google Scholar
  10. 15.
    Berens, H., Xu, Y.: On Bernstein-Durrmeyer polynomials with Jacobi weights. In: Chui, C.K. (ed.) Approximation Theory and Functionals Analysis, pp. 25–46. Academic Press, Boston (1991)Google Scholar
  11. 19.
    Bernstein, S.N.: Complétement a l’article de E. Voronowskaja. C.R. Acad. Sci. U.R.S.S. Ser. A. 4, 86–92 (1932)Google Scholar
  12. 20.
    Bernstein, S.N.: Leo̧ns sur les Propriétés Extrémales et la Meilleure Approximations des Fonctions Analytiques d’Une Variable Réelle. Gauthier-Villars, Paris (1926)Google Scholar
  13. 21.
    Bleimann, G, Butzer, P.L., Hahn, L.: A Bernstein-type operator approximating continuous functions on the semi-axis. Indag. Math. 42, 255–262 (1980)MathSciNetMATHGoogle Scholar
  14. 24.
    Chen, W.Z.: On the modified Bernstein-Durrmeyer operators. In: Report of the Fifth Chinese Conference on Approximation Theory, Zhen Zhou, China (1987)Google Scholar
  15. 26.
    Cimoca, G., Lupaş, A.: Two generalizations of the Meyer-König and Zeller operator. Mathematica(Cluj) 9(32)(2), 233–240 (1967)Google Scholar
  16. 27.
    Coleman, J.P., Smith, R.A.: The Faber polynomials for circular sectors. Math. Comput. 49, 231–241 (1987)MathSciNetMATHCrossRefGoogle Scholar
  17. 28.
    Coleman, J.P., Myers, N.J.: The Faber polynomials for annular sectors. Math. Comput. 64, 181–203 (1995)MathSciNetMATHCrossRefGoogle Scholar
  18. 29.
    Curtiss, J.H.: Faber polynomials and the Faber series. Am. Math. Month 78(6), 5677–596 (1971)MathSciNetCrossRefGoogle Scholar
  19. 33.
    Dzjadyk, V.K.: Introduction to the Theory of Uniform Approximation of Functions by Polynomials (Russian). Nauka, Moscow (1977)Google Scholar
  20. 34.
    Faber, G.: Über polynomische Entwicklungen. Math. Ann. 57, 398–408 (1903)Google Scholar
  21. 38.
    Gaier, D.: Lectures on Complex Approximation. Birkhauser, Boston (1987)MATHCrossRefGoogle Scholar
  22. 39.
    Gal, S.G.: Shape Preserving Approximation by Real and Complex Polynomials. Birkhauser, Boston, Basel, Berlin (2008)MATHCrossRefGoogle Scholar
  23. 41.
    Gal, S.G.: Voronovskaja’s theorem and iterations for complex Bernstein polynomials in compact disks. Mediterr. J. Math. 5(3), 253–272 (2008)MathSciNetMATHCrossRefGoogle Scholar
  24. 43.
    Gal, S.G.: Exact orders in simultaneous approximation by complex Bernstein polynomials. J. Concr. Appl. Math. 7(3), 215–220 (2009)MathSciNetMATHGoogle Scholar
  25. 44.
    Gal, S.G.: Approximation by complex Bernstein-Stancu polynomials in compact disks. Results Math. 53(3–4), 245–256 (2009)MathSciNetMATHCrossRefGoogle Scholar
  26. 45.
    Gal, S.G.: Exact orders in simultaneous approximation by complex Bernstein-Stancu polynomials. Revue d’Anal. Numér. Théor. de L’Approx. (Cluj-Napoca) 37(1), 47–52 (2008)Google Scholar
  27. 46.
    Gal, S.G.: Generalized Voronovskaja’s theorem and approximation by Butzer’s combinations of complex Bernstein polynomials. Results Math. 53(3–4), 257–268 (2009)MathSciNetMATHCrossRefGoogle Scholar
  28. 47.
    Gal, S.G.: Approximation by complex Bernstein-Kantorovich and Stancu-Kantorovich polynomials and their iterates in compact disks. Revue D’Anal. Numér. Théor. de L’Approx. (Cluj) 37(2), 159–168 (2008)Google Scholar
  29. 49.
    Gal, S.G.: Approximation by Complex Bernstein and Convolution Type Operators. World Scientific Publishing, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai (2009)MATHGoogle Scholar
  30. 50.
    Gal, S.G.: Approximation by complex genuine Durrmeyer type polynomials in compact disks. Appl. Math. Comput. 217, 1913–1920 (2010)MathSciNetMATHCrossRefGoogle Scholar
  31. 51.
    Gal, S.G.: Approximation by complex Bernstein-Durrmeyer polynomials with Jacobi weightds in compact disks. Mathematica Balkanica (N.S.) 24(1–2), 103–119 (2010)Google Scholar
  32. 52.
    Gal, S.G.: Approximation by complex Lorentz polynomials. Math. Comm. 16(2011), 65–75 (2011)MathSciNetGoogle Scholar
  33. 53.
    Gal, S.G.: Differentiated generalized Voronovskaja’s theorem in compact disks. Results Math. 61(3), 247–253 (2012)MathSciNetCrossRefGoogle Scholar
  34. 58.
    Gal, S.G.: Approximation by complex q-Lorentz polynomials, q > 1. Mathematica (Cluj) 54(77)(1), 53–63 (2012)Google Scholar
  35. 59.
    Gal, S.G.: Approximation of analytic functions without exponential growth conditions by complex Favard-Szász-Mirakjan operators. Rendiconti del Circolo Matematico di Palermo 59(3), 367–376 (2010)MathSciNetMATHCrossRefGoogle Scholar
  36. 60.
    Gal, S.G.: Approximation by quaternion q-Bernstein polynomials, q > 1. Adv. Appl. Clifford Alg. 22(2), 313–319 (2012)MathSciNetMATHCrossRefGoogle Scholar
  37. 61.
    Gal, S.G.: Approximation in compact sets by q-Stancu-Faber polynomials, q > 1. Comput. Math. Appl. 61(10), 3003–3009 (2011)MathSciNetMATHCrossRefGoogle Scholar
  38. 62.
    Gal, S.G.: Voronovskaja-type results in compact disks for quaternion q-Bernstein operators, q ≥ 1. Complex Anal. Oper. Theory 6(2), 515–527 (2012)MathSciNetCrossRefGoogle Scholar
  39. 63.
    Gal, S.G.: (Online access) Erratum to: Differentiated generalized Voronovskaja’s theorem in compact disks. Results Math. DOI 10.1007/s00025-012-0295-1, published online 23 October 2012Google Scholar
  40. 66.
    Gal, S.G., Gupta, V.: (Online access) Approximation by complex Beta operators of first kind in strips of compact disks. Mediterranean J. Math. 10(1), 31–39 (2013)Google Scholar
  41. 67.
    Gal, S.G., Gupta, V., Mahmudov, N.I.: Approximation by a complex q-Durrmeyer type operator. Ann. Univ. Ferrara 58, 65–87 (2012)MathSciNetCrossRefGoogle Scholar
  42. 68.
    Gal, S.G., Mahmudov, N.I., Kara, M.: (Online access) Approximation by complex q-Szàsz-Kantorovich operators in compact disks. Complex Anal. Oper. Theory, DOI: 10.1007/s11785-012-0257-3Google Scholar
  43. 69.
    Gentili, G., Stoppato, C.: Power series and analyticity over the quaternions. Math. Ann. 352(1), 113–131 (2012)MathSciNetMATHCrossRefGoogle Scholar
  44. 70.
    Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternionic variable. Advances Math. 216, 279–301 (2007)MathSciNetMATHCrossRefGoogle Scholar
  45. 71.
    Gonska, H., Piţul, P., Raşa, I.: On Peano’s form of the Taylor remainder, Voronovskaja’s theorem and the commutator of positive linear operators. In: Proceed. Intern. Conf. on “Numer. Anal., Approx. Theory”, NAAT, Cluj-Napoca, Casa Cartii de Stiinta, Cluj-Napoca, pp. 55–80, 2006Google Scholar
  46. 72.
    Gonska H., Raşa, I.: Asymptotic behaviour of differentiated Bernstein polynomials. Mat. Vesnik, 61, 53–60 (2009)MathSciNetMATHGoogle Scholar
  47. 73.
    Goodman, T.N.T., Sharma, A.: A modified Bernstein-Schoenberg operator. In: Sendov, Bl. et al (eds.) Constructive Theory of Functions - Varna 1987, pp. 166–173. Bulgar. Acad. Sci., Sofia (1988)Google Scholar
  48. 74.
    Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions, Pure and Applied Mathematics, vol. 255. Marcel Dekker, New York (2003)Google Scholar
  49. 75.
    Gupta, V.: Some approximation properties of q-Durrmeyer type operators. Appl. Math. Comput. 197(1), 172–178 (2008)MathSciNetMATHCrossRefGoogle Scholar
  50. 76.
    Gupta, V., Finta, Z.: On certain q-Durrmeyer type operators. Appl. Math. Comput. 209(2), 415–420 (2009)MathSciNetMATHCrossRefGoogle Scholar
  51. 77.
    Hasson, M.: Expansion of analytic functions of an operator in series of Faber polynomials. Bull. Aust. Math. Soc. 56, 303–318 (1997)MathSciNetMATHCrossRefGoogle Scholar
  52. 78.
    He, M.: Explicit representations of Faber polynomials for m-cusped hypocycloids. J. Approx. Theory 87, 137–147 (1996)MathSciNetMATHCrossRefGoogle Scholar
  53. 79.
    He, M.: The Faber polynomials for m-fold symmetric domains. J. Comput. Appl. Math. 54, 313–324 (1994)MathSciNetMATHCrossRefGoogle Scholar
  54. 80.
    He, M.: The Faber polynomials for circular lunes. Comput. Math. Appl. 30, 307–315 (1995)MathSciNetMATHCrossRefGoogle Scholar
  55. 81.
    Henrici, P.: Applied and Computational Analysis, vol. I. Wiley, New York (1974)MATHGoogle Scholar
  56. 82.
    He, M., Saff, E.B.: The zeros of Faber polynomials for and m-cusped hypocycloid. J. Approx. Theory 78, 410–432 (1994)MathSciNetMATHCrossRefGoogle Scholar
  57. 87.
    Jackson, F.H.: On q-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)MATHGoogle Scholar
  58. 90.
    Khan, M.K.: Approximation properties of Beta operators. In: Progress in Approximation Theory, pp. 483–495. Academic Press, New York (1991)Google Scholar
  59. 91.
    Kohr, G., Mocanu, P.T.: Special Chapters of Complex Analysis (in Romanian). University Press, Cluj-Napoca (2005)Google Scholar
  60. 95.
    Leviatan, D.: On the remainder in the approximation of functions by Bernstein-type Operators. J. Appox. Theory 2, 400–409 (1969)MathSciNetMATHCrossRefGoogle Scholar
  61. 96.
    Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea Publication, New York (1986)MATHGoogle Scholar
  62. 97.
    Lorentz, G.G.: Approximation of Functions. Chelsea Publication, New York (1987)Google Scholar
  63. 99.
    Lupaş, A.: On Bernstein power series. Mathematica(Cluj) 8(31), 287–296 (1966)Google Scholar
  64. 100.
    Lupaş, A.: Some properties of the linear positive operators, III. Revue d’Analyse Numer. Théor. Approx. 3, 47–61 (1974)MATHGoogle Scholar
  65. 101.
    Lupas, A.: Die Folge der Beta-Operatoren. Dissertation, Univ. Stuttgart, Stuttgart (1972)Google Scholar
  66. 102.
    Lupaş, L., Lupaş, A.: Polynomials of binomial type and approximation operators. Stud. Univ. “Babes-Bolyia”, Math. 32(4), 60–69 (1987)Google Scholar
  67. 103.
    Lupaş, A., Müller, M.: Approximationseigenschaften der Gammaoperatoren. Math. Zeitschr. 98, 208–226 (1967)MATHCrossRefGoogle Scholar
  68. 104.
    Mahmudov, N.I.: Convergence properties and iterations for q-Stancu polynomials in compact disks. Comput. Math. Appl. 59(12), 3763–3769 (2010)MathSciNetMATHCrossRefGoogle Scholar
  69. 105.
    Mahmudov, N.I.: Approximation properties of complex q-Szász-Mirakjan operators in compact disks. Comput. Math. Appl. 60, 1784–1791 (2010)MathSciNetMATHCrossRefGoogle Scholar
  70. 106.
    Mahmudov, N.I., Kara, M.: Approximation theorems for generalized complex Kantorovich-type operators. J. Appl. Math. 2012, Article ID 454579, 14 pages (2012). Doi:10.1155/2012/454579Google Scholar
  71. 108.
    Mejlihzon, A.Z.: On the notion of monogenic quaternions (in Russian). Dokl. Akad. Nauk SSSR 59, 431–434 (1948).Google Scholar
  72. 109.
    Meyer-König, W., Zeller, K.: Bernsteinsche Potenzreihen. Studia Math. 19, 89–94 (1960)MATHGoogle Scholar
  73. 110.
    Mocanu, P.T., Bulboacă, T., Sălăgean, Gr. St.: Geometric Function Theory of Univalent Functions, (in Romanian). Science Book’s House, Cluj-Napoca (1999)Google Scholar
  74. 111.
    Mocică, G.: Problems of Special Functions (in Romanian). Edit. Didact. Pedag., Bucharest (1988)Google Scholar
  75. 112.
    Moisil, Gr.C.: Sur les quaternions monogènes. Bull. Sci. Math. (Paris) LV, 168–174 (1931)Google Scholar
  76. 113.
    Moldovan, G.: Discrete convolutions for functions of several variables and linear positive operators (Romanian). Stud. Univ. “Babes-Bolyai” Ser. Math. 19(1), 51–57 (1974)Google Scholar
  77. 114.
    Mühlbach, G.: Verallgemeinerungen der Bernstein - und der Lagrangepolynome. Rev. Roumaine Math. Pures Appl. 15(8), 1235–1252 (1970)MathSciNetMATHGoogle Scholar
  78. 117.
    Ostrovska, S.: On the q-Bernstein polynomials and their iterates. Adv. Stud. Contemp. Math. 11, 193–204 (2005)MathSciNetMATHGoogle Scholar
  79. 118.
    Ostrovska, S.: q-Bernstein polynomials and their iterates. J. Approx. Theory 123, 232–255 (2003)MathSciNetMATHGoogle Scholar
  80. 119.
    Parvanov, P.P., Popov, B.D.: The limit case of Bernstein’s operators with Jacobi weights. Math. Balkanica, N.S. 8, 165–177 (1994)Google Scholar
  81. 120.
    Păltănea, R.: Sur une opérateur polynomial defini sur l’ensemble des fonctions intégrables. In: Itinerant Seminar on Functional Equations, Approximation and Convexity, (Cluj-Napoca), Preprint 83-2, Univ. “Babes-Bolyai”, Cluj-Napoca, pp. 101–106 (1983)Google Scholar
  82. 121.
    Păltănea, R.: Inverse theorem for a polynomial operator. In: Itinerant Seminar on Functional Equations, Aproximation and Convexity (Cluj-Napoca), Preprint 85-6, University “Babes-Bolyai”, Cluj-Napoca, pp. 149–152 (1985)Google Scholar
  83. 122.
    Păltănea, R.: Une classe générale d’operateur polynomiaux. Revue Anal. Numér. Théor. Approx. (Cluj) 17(1), 49–52 (1988)Google Scholar
  84. 123.
    Phillips, G.M.: Bernstein polynomials based on the q-integers. Ann. Numer. Math. 4, 511–518 (1997)MathSciNetMATHGoogle Scholar
  85. 124.
    Radon, J.: Über die Randwertaufgaben beim logarithmischen Potential. Sitz.-Ber. Wien Akad. Wiss. Abt. IIa 128, 1123–1167 (1919)Google Scholar
  86. 126.
    Raşa, I.: On Soardi’s Bernstein operators of second kind. In: Lupsa, L., Ivan, M. (eds.) Proceed. Conf. for Analysis, Functional Equations, Approximation and Convexity, pp. 264–271. Carpatica Press, Cluj-Napoca (1999)Google Scholar
  87. 128.
    Sauer, T.: The genuine Bernstein-Durrmeyer operator on a simplex. Results Math. 26(1–2), 99–130 (1994)MathSciNetMATHCrossRefGoogle Scholar
  88. 130.
    Schurer, F.: Linear positive operators in approximation theory. Math. Inst. Tech. Univ. Delft Report, (1962)Google Scholar
  89. 132.
    Soardi, P.: Bernstein polynomials and random walks on hypergroups. In: Herbert, H. (ed.) Proceedings of the 10th Oberwolfach Conference on Probability Measures on Groups, X, 1990, pp. 387–393. Plenum Publication, New York (1991)Google Scholar
  90. 134.
    Stancu, D.D.: Approximation of functions by means of some new classes of linear polynomial operators. In: Colatz, L., Meinardus, G. (eds.) Proc. Conf. Math. Res. Inst. Oberwolfach Numerische Methoden der Approximationstheorie, 1971, pp. 187–203. Birkhäuser, Basel (1972)Google Scholar
  91. 135.
    Stepanets, A.I.: Classification and Approximation of Periodic Functions, Mathematics and Its Applications, vol. 333. Kluwer Academic, Dordrecht, Boston, London (1995)CrossRefGoogle Scholar
  92. 136.
    Suetin, P.K.: Series of Faber Polynomials. Gordon and Breach, Amsterdam (1998)MATHGoogle Scholar
  93. 142.
    Voronovskaja, E.V.: Determination de la forme asymptotique de l’approximation des fonctions par les polynômes de M. Bernstein (in Russian). C.R. (Dokl.) Acad. Sci. U.R.S.S. A 4, 79–85 (1932)Google Scholar
  94. 145.
    Wang, H., Wu, X.Z.: Saturation of convergence for q-Bernstein polynomials in the case q ≥ 1. J. Math. Anal. Appl. 337, 744–750 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Sorin G. Gal
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of OradeaOradeaRomania

Personalised recommendations