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Overconvergence in ℂ of Some Bernstein-Type Operators

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Abstract

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

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References

  1. Abel, U., Gupta, V., Mohapatra, R.N.: Local approximation by beta operators. Nonlinear Anal. 62(1), 41–52 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abel, U., Heilmann, H.: The complete asymptotic expansion for Bernstein-Durrmeyer operator with Jacobi weights. Mediterr. J. Math. 1(4), 487–499 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alexander, J.W.: Functions which map the interior of the unit circle upon simple regions. Ann. Math. 17(1915), 12–22 (1915)

    Article  MATH  Google Scholar 

  5. Altomare, F., Mangino, E.: On a generalization of Baskakov operator. Rev. Roumaine Math. Pures Appl. 44(5–6), 683–705 (1999)

    MathSciNet  MATH  Google Scholar 

  6. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  9. Bartolomeu, J., He, M.: On Faber polynomials generated by an m-star. Math. Comput. 62, 277–288 (1994)

    Google Scholar 

  10. 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. 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. 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. Bleimann, G, Butzer, P.L., Hahn, L.: A Bernstein-type operator approximating continuous functions on the semi-axis. Indag. Math. 42, 255–262 (1980)

    MathSciNet  MATH  Google Scholar 

  14. 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. Cimoca, G., Lupaş, A.: Two generalizations of the Meyer-König and Zeller operator. Mathematica(Cluj) 9(32)(2), 233–240 (1967)

    Google Scholar 

  16. Coleman, J.P., Smith, R.A.: The Faber polynomials for circular sectors. Math. Comput. 49, 231–241 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Coleman, J.P., Myers, N.J.: The Faber polynomials for annular sectors. Math. Comput. 64, 181–203 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Curtiss, J.H.: Faber polynomials and the Faber series. Am. Math. Month 78(6), 5677–596 (1971)

    Article  MathSciNet  Google Scholar 

  19. Dzjadyk, V.K.: Introduction to the Theory of Uniform Approximation of Functions by Polynomials (Russian). Nauka, Moscow (1977)

    Google Scholar 

  20. Faber, G.: Über polynomische Entwicklungen. Math. Ann. 57, 398–408 (1903)

    Google Scholar 

  21. Gaier, D.: Lectures on Complex Approximation. Birkhauser, Boston (1987)

    Book  MATH  Google Scholar 

  22. Gal, S.G.: Shape Preserving Approximation by Real and Complex Polynomials. Birkhauser, Boston, Basel, Berlin (2008)

    Book  MATH  Google Scholar 

  23. Gal, S.G.: Voronovskaja’s theorem and iterations for complex Bernstein polynomials in compact disks. Mediterr. J. Math. 5(3), 253–272 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gal, S.G.: Exact orders in simultaneous approximation by complex Bernstein polynomials. J. Concr. Appl. Math. 7(3), 215–220 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Gal, S.G.: Approximation by complex Bernstein-Stancu polynomials in compact disks. Results Math. 53(3–4), 245–256 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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. 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)

    MATH  Google Scholar 

  30. Gal, S.G.: Approximation by complex genuine Durrmeyer type polynomials in compact disks. Appl. Math. Comput. 217, 1913–1920 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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. Gal, S.G.: Approximation by complex Lorentz polynomials. Math. Comm. 16(2011), 65–75 (2011)

    MathSciNet  Google Scholar 

  33. Gal, S.G.: Differentiated generalized Voronovskaja’s theorem in compact disks. Results Math. 61(3), 247–253 (2012)

    Article  MathSciNet  Google Scholar 

  34. Gal, S.G.: Approximation by complex q-Lorentz polynomials, q > 1. Mathematica (Cluj) 54(77)(1), 53–63 (2012)

    Google Scholar 

  35. 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)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gal, S.G.: Approximation by quaternion q-Bernstein polynomials, q > 1. Adv. Appl. Clifford Alg. 22(2), 313–319 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Gal, S.G.: Approximation in compact sets by q-Stancu-Faber polynomials, q > 1. Comput. Math. Appl. 61(10), 3003–3009 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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)

    Article  MathSciNet  Google Scholar 

  39. 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 2012

    Google Scholar 

  40. 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. Gal, S.G., Gupta, V., Mahmudov, N.I.: Approximation by a complex q-Durrmeyer type operator. Ann. Univ. Ferrara 58, 65–87 (2012)

    Article  MathSciNet  Google Scholar 

  42. 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/s11785-012-0257-3

    Google Scholar 

  43. Gentili, G., Stoppato, C.: Power series and analyticity over the quaternions. Math. Ann. 352(1), 113–131 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternionic variable. Advances Math. 216, 279–301 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  45. 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, 2006

    Google Scholar 

  46. Gonska H., Raşa, I.: Asymptotic behaviour of differentiated Bernstein polynomials. Mat. Vesnik, 61, 53–60 (2009)

    MathSciNet  MATH  Google Scholar 

  47. 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. 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. Gupta, V.: Some approximation properties of q-Durrmeyer type operators. Appl. Math. Comput. 197(1), 172–178 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. Gupta, V., Finta, Z.: On certain q-Durrmeyer type operators. Appl. Math. Comput. 209(2), 415–420 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. Hasson, M.: Expansion of analytic functions of an operator in series of Faber polynomials. Bull. Aust. Math. Soc. 56, 303–318 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  52. He, M.: Explicit representations of Faber polynomials for m-cusped hypocycloids. J. Approx. Theory 87, 137–147 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  53. He, M.: The Faber polynomials for m-fold symmetric domains. J. Comput. Appl. Math. 54, 313–324 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  54. He, M.: The Faber polynomials for circular lunes. Comput. Math. Appl. 30, 307–315 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  55. Henrici, P.: Applied and Computational Analysis, vol. I. Wiley, New York (1974)

    MATH  Google Scholar 

  56. He, M., Saff, E.B.: The zeros of Faber polynomials for and m-cusped hypocycloid. J. Approx. Theory 78, 410–432 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  57. Jackson, F.H.: On q-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)

    MATH  Google Scholar 

  58. Khan, M.K.: Approximation properties of Beta operators. In: Progress in Approximation Theory, pp. 483–495. Academic Press, New York (1991)

    Google Scholar 

  59. Kohr, G., Mocanu, P.T.: Special Chapters of Complex Analysis (in Romanian). University Press, Cluj-Napoca (2005)

    Google Scholar 

  60. Leviatan, D.: On the remainder in the approximation of functions by Bernstein-type Operators. J. Appox. Theory 2, 400–409 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  61. Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea Publication, New York (1986)

    MATH  Google Scholar 

  62. Lorentz, G.G.: Approximation of Functions. Chelsea Publication, New York (1987)

    Google Scholar 

  63. Lupaş, A.: On Bernstein power series. Mathematica(Cluj) 8(31), 287–296 (1966)

    Google Scholar 

  64. Lupaş, A.: Some properties of the linear positive operators, III. Revue d’Analyse Numer. Théor. Approx. 3, 47–61 (1974)

    MATH  Google Scholar 

  65. Lupas, A.: Die Folge der Beta-Operatoren. Dissertation, Univ. Stuttgart, Stuttgart (1972)

    Google Scholar 

  66. Lupaş, L., Lupaş, A.: Polynomials of binomial type and approximation operators. Stud. Univ. “Babes-Bolyia”, Math. 32(4), 60–69 (1987)

    Google Scholar 

  67. Lupaş, A., Müller, M.: Approximationseigenschaften der Gammaoperatoren. Math. Zeitschr. 98, 208–226 (1967)

    Article  MATH  Google Scholar 

  68. Mahmudov, N.I.: Convergence properties and iterations for q-Stancu polynomials in compact disks. Comput. Math. Appl. 59(12), 3763–3769 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  69. Mahmudov, N.I.: Approximation properties of complex q-Szász-Mirakjan operators in compact disks. Comput. Math. Appl. 60, 1784–1791 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  70. Mahmudov, N.I., Kara, M.: Approximation theorems for generalized complex Kantorovich-type operators. J. Appl. Math. 2012, Article ID 454579, 14 pages (2012). Doi:10.1155/2012/454579

    Google Scholar 

  71. Mejlihzon, A.Z.: On the notion of monogenic quaternions (in Russian). Dokl. Akad. Nauk SSSR 59, 431–434 (1948).

    Google Scholar 

  72. Meyer-König, W., Zeller, K.: Bernsteinsche Potenzreihen. Studia Math. 19, 89–94 (1960)

    MATH  Google Scholar 

  73. 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. Mocică, G.: Problems of Special Functions (in Romanian). Edit. Didact. Pedag., Bucharest (1988)

    Google Scholar 

  75. Moisil, Gr.C.: Sur les quaternions monogènes. Bull. Sci. Math. (Paris) LV, 168–174 (1931)

    Google Scholar 

  76. 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. Mühlbach, G.: Verallgemeinerungen der Bernstein - und der Lagrangepolynome. Rev. Roumaine Math. Pures Appl. 15(8), 1235–1252 (1970)

    MathSciNet  MATH  Google Scholar 

  78. Ostrovska, S.: On the q-Bernstein polynomials and their iterates. Adv. Stud. Contemp. Math. 11, 193–204 (2005)

    MathSciNet  MATH  Google Scholar 

  79. Ostrovska, S.: q-Bernstein polynomials and their iterates. J. Approx. Theory 123, 232–255 (2003)

    MathSciNet  MATH  Google Scholar 

  80. 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. 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. 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. Păltănea, R.: Une classe générale d’operateur polynomiaux. Revue Anal. Numér. Théor. Approx. (Cluj) 17(1), 49–52 (1988)

    Google Scholar 

  84. Phillips, G.M.: Bernstein polynomials based on the q-integers. Ann. Numer. Math. 4, 511–518 (1997)

    MathSciNet  MATH  Google Scholar 

  85. Radon, J.: Über die Randwertaufgaben beim logarithmischen Potential. Sitz.-Ber. Wien Akad. Wiss. Abt. IIa 128, 1123–1167 (1919)

    Google Scholar 

  86. 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. Sauer, T.: The genuine Bernstein-Durrmeyer operator on a simplex. Results Math. 26(1–2), 99–130 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  88. Schurer, F.: Linear positive operators in approximation theory. Math. Inst. Tech. Univ. Delft Report, (1962)

    Google Scholar 

  89. 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. 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. Stepanets, A.I.: Classification and Approximation of Periodic Functions, Mathematics and Its Applications, vol. 333. Kluwer Academic, Dordrecht, Boston, London (1995)

    Book  Google Scholar 

  92. Suetin, P.K.: Series of Faber Polynomials. Gordon and Breach, Amsterdam (1998)

    MATH  Google Scholar 

  93. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

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Gal, S.G. (2013). Overconvergence in ℂ of Some Bernstein-Type Operators. In: Overconvergence in Complex Approximation. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7098-4_1

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