Skip to main content
SpringerLink
Log in

The q-Versions of the Bernstein Operator: From Mere Analogies to Further Developments

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The article exhibits a review of results on two popular q-versions of the Bernstein polynomials, namely, the Lupaş q-analogue and the q-Bernstein polynomials. Their similarities and distinctions are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agratini O.: On certain q-analogues of the Bernstein operators. Carpathian J. Math. 24(3), 281–286 (2009)

    MATH  Google Scholar 

  2. Andrews G.E., Askey R., Roy R.: Special Functions. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  3. Barbosu D.: Some generalized bivariate Bernstein operators. Math. Notes (Miskolc) 1(1), 3–10 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Bernstein, S.N.: Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Communic. Soc. Math. Charkow. série 2 13, 1–2 (1912)

  5. Boehm W., Müller A.: On de Casteljau’s algorithm. Comput. Aided Geom. Des. 16, 587–605 (1999)

    Article  MATH  Google Scholar 

  6. Bohner, M., Guseinov, G., Peterson, A.: Introduction to the time scales calculus. In: Bohner, M., Peterson, A. (eds.) Advances in Dynamic Equations on Time Scales, pp. 1–15. Birkhäuser Boston, Inc., Boston (2003)

  7. Charalambides Ch.A.: The q-Bernstein basis as a q-binomial distribution. J. Stat. Plan. Inference 140(8), 2184–2190 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cooper S., Waldron S.: The Eigenstructure of the Bernstein operator. J. Approx. Theory. 105, 133–165 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Delgado J., Pena J.M.: Accurate computations with collocation matrices of q-Bernstein polynomials. SIAM J. Matrix Anal. Appl. 36(2), 880–893 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Derriennic, M.-M.: Modified Bernstein polynomials and Jacobi polynomials in q-calculus. Rendiconti Del Circolo Matematico Di Palermo. Serie II, Suppl. 76, 269–290 (2005)

  11. Dikmen, A.B., Lukashov, A.: Generating functions method for classical positive operators, their q-analogues and generalizations. Positivity (2015). doi:10.1007/s11117-015-0362-4

  12. Finta Z.: Note on a Korovkin-type theorem. J. Math. Anal. Appl. 415(2), 750–759 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gal S.G.: Voronovskaja’s theorem, shape-preserving properties and iterations for complex q-Bernstein polynomials. Studia Scientiarum Mathematicarum Hungarica 48(1), 23–43 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Gal, S.G.: Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics, vol. 8. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2009)

  16. Gupta V., Wang H.: The rate of convergence of q-Durrmeyer operators for 0 < q < 1. Math. Methods Appl. Sci. 31(16), 1946–1955 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Han L.-W., Chu Y., Qiu Z.-Y.: Generalized Bezier curves and surfaces based on Lupas q-analogue of Bernstein operator. J. Comput. Appl. Math. 261, 352–363 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hardy, G.H.: Divergent Series. Oxford University Press, Oxford (1949)

  19. Il’inskii A.: A probabilistic approach to q-polynomial coefficients, Euler and Stirling numbers I. Mat. Fiz. Anal. Geom. 11(4), 434–448 (2004)

    MathSciNet  MATH  Google Scholar 

  20. Il’inskii A., Ostrovska S.: Convergence of generalized Bernstein polynomials. J. Approx. Theory 116, 100–112 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jing S.: The q-deformed binomial distribution and its asymptotic behaviour. J. Phys. A Math. Gen. 27, 493–499 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kantorovich L.V.: La representation explicite d’une fonction mesurable arbitraire dans la forme de la limite d’une suite de polynomes. Math. Sbornik. 41, 508–510 (1934)

    Google Scholar 

  23. Levin, B.Y.: Lectures on Entire Functions. Translations of Mathematical Monographs, vol. 150. American Mathematical Society, Providence, RI (1996)

  24. Lewanowicz S., Woźny P.: Generalized Bernstein polynomials. BIT 44(1), 63–78 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lorentz G.G.: Bernstein Polynomials. Chelsea, New York (1986)

    MATH  Google Scholar 

  26. Lupaş, A.: A q-analogue of the Bernstein operator. University of Cluj-Napoca. Seminar on numerical and statistical calculus, vol. 9 (1987)

  27. Mahmudov N.: The moments for q-Bernstein operators in the case 0 < q < 1. Numer. Algorithms 53, 439–450 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mahmudov N., Sabancigil P.: A q-analogue of the Meyer-Konig and Zeller operators. Bull. Malays. Math. Sci. Soc. 35(1), 39–51 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Mahmudov N.: Korovkin-type theorems and applications. Cent. Eur. J. Math. 7(2), 348–356 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Mahmudov N.: Asymptotic properties of powers of linear positive operators which preserve e 2. Comput. Math. Appl. 62(12), 4568–4575 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mahmudov N.: Approximation properties of bivariate complex q-Bernstein polynomials in the case q >  1. Czechoslov. Math. J. 62(2), 557–566 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mahmudov N.: Approximation by q-Durrmeyer type polynomials in compact disks in the case q >  1. Appl. Math. Comput. 237, 293–303 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Mahmudov N., Sabancigil P.: Voronovskaja type theorem for the Lupas q-analogue of the Bernstein operators. Math. Commun. 17(1), 83–91 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Ostrovska S.: On the improvement of analytic properties under the limit q-Bernstein operator. J. Approx. Theory 138, 37–53 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ostrovska S.: On the Lupaş q-analogue of the Bernstein operator. Rocky Mt. J. Math. 36(5), 1615–1629 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ostrovska S.: Positive linear operators generated by analytic functions. Proc. Indian Acad. Sci. (Math. Sci.) 117(4), 485–493 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ostrovska, S.: The functional-analytic properties of the limit q-Bernstein operator. J. Funct. Spaces Appl. (2012). doi:10.1155/2012/280314

  39. Ostrovska S.: On the approximation of analytic functions by the q-Bernstein polynomials in the case q >  1. Electron. Trans. Numer. Anal. 37, 105–112 (2010)

    MathSciNet  MATH  Google Scholar 

  40. Ostrovska S.: On the Lupaş q-transform. Comput. Math. Appl. 61, 527–532 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ostrovska S.: Analytical properties of the Lupaş q-transform. J. Math. Anal. Appl. 394, 177–185 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ostrovska S.: Geometric properties of the Lupaş q-transform. Banach J. Math. Anal. 8(2), 139–145 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ostrovska, S.: Functions whose smoothness is not improved under the limit q-Bernstein operator. J. Inequal. Appl. 2012, 297 (2012). doi:10.1186/1029-242X-2012-297

  44. Ostrovska, S., Özban, A.Y.: On the q-Bernstein polynomials of unbounded functions with q > 1. Abstr. Appl. Anal. (2013). Article ID 349156

  45. Ostrovska S., A.Y.: On the q-Bernstein polynomials of rational functions with real poles. J. Math. Anal. Appl. 413, 547–556 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ostrovska S., Turan A.Y., Turan M.: How do singularities of functions affect the convergence of q-Bernstein polynomials?. J. Math. Inequal. 9(1), 121–136 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ostrovska S., Turan M.: On the eigenvectors of the q-Bernstein operators. Math. Methods Appl. Sci. 37(4), 562–570 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Ostrovskii I., Ostrovska S.: On the analyticity of functions approximated by their q-Bernstein polynomials when q >  1. Appl. Math. Comput. 217(1), 65–72 (2010)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  50. Phillips G.M.: Interpolation and Approximation by Polynomials. Springer-Verlag, New York (2003)

    Book  MATH  Google Scholar 

  51. Phillips G.M.: A survey of results on the q-Bernstein polynomials. IMA J. Numer. Anal. 30, 277–288 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  52. Ren M.-Y., Zeng X.-M.: Approximation by complex q-Bernstein–Schurer operators in compact disks. Georgian Math. J. 20(2), 377–395 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. Trif T.: Meyer-König and Zeller operators based on the q-integers. Rev. Anal. Numér. Théor. Approx. 29(2), 221–229 (2000)

    MathSciNet  MATH  Google Scholar 

  54. Videnskii, V.S.: Bernstein Polynomials. Leningrad State Pedagogical University, Leningrad (1990) (Russian)

  55. Videnskii, V.S.: On q-Bernstein polynomials and related positive linear operators. In: Some current problems in modern mathematics and education in mathematics, pp.118–126. Ross. Gos. Ped. Univ., St. Petersburg (2004) (Russian)

  56. Videnskii V.S.: On some classes of q-parametric positive operators. Oper. Theory Adv. Appl. 158, 213–222 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  57. Videnskii, V.S.: A remark on the rational linear operators considered by A. Lupas. In: Some current problems in modern mathematics and education in mathematics, pp. 134–146. Ross. Gos. Ped. Univ., St. Petersburg (2008) (Russian)

  58. Videnskii, V.S.: On the centenary of the discovery of Bernstein polynomials. Some current problems in modern mathematics and education in mathematics, pp. 5–11. Izdat. RGPU im. A. I. Gertsena, St. Petersburg (2013) (Russian)

  59. Videnskii V.S.: Papers of L. V. Kantorovich on Bernstein polynomials. Vestnik St. Petersburg Univ. Math. 46(2), 85–88 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  60. Wang H.: Korovkin-type theorem and application. J. Approx. Theory 132(2), 258–264 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  61. Wang H.: Properties of convergence for the q-Meyer-Konig and Zeller operators. J. Math. Anal. Appl. 335(2), 1360–1373 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  62. Wang H.: Properties of convergence for ω, q -Bernstein polynomials. J. Math. Anal. Appl. 340(2), 1096–1108 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  63. Wang, H., Zhang, Y.: The rate of convergence of Lupas q-analogue of the Bernstein operators. Abstr. Appl. Anal. (2014). doi:10.1155/2014/521709

  64. Wang H., Meng F.: The rate of convergence of q-Bernstein polynomials for 0 < q < 1. J. Approx. Theory 136(2), 151–158 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  65. Wang H.: Voronovskaya type formulas and saturation of convergence for q-Bernstein polynomials for 0 < q < 1. J. Approx. Theory 145, 182–195 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  66. Wang H., Ostrovska S.: The norm estimates for the q-Bernstein operator in the case q > 1. Math. Comput. 79(269), 353–363 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  67. 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 

  68. Weba M.: On the approximation of unbounded functions by Bernstein polynomials. East J. Approx. 9(3), 351–356 (2003)

    MathSciNet  MATH  Google Scholar 

  69. Wu Z.: The saturation of convergence on the interval [0,1] for the q-Bernstein polynomials in the case q > 1. J. Math. Anal. Appl. 357, 137–141 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  70. Wu, X.Z.: Approximation by q-Bernstein polynomials in the case q→ 1+. Abstr. Appl. Anal. (2014). doi:10.1155/2014/259491

  71. Xiang X., He Q., Yang W.: Convergence rate for iterates of q-Bernstein polynomials. Anal. Theory Appl. 23(3), 243–254 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  72. Zeng J., Zhang C.: A q-analog of Newton’s series, Stirling functions and Eulerian functions. Results Math. 25(3–4), 370–391 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Atilim University, Incek, 06836, Ankara, Turkey

    Sofiya Ostrovska

Authors
  1. Sofiya Ostrovska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sofiya Ostrovska.

Additional information

Dedicated to the memory of Professor Viktor Solomonovich Videnskii (1922–2015)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ostrovska, S. The q-Versions of the Bernstein Operator: From Mere Analogies to Further Developments. Results. Math. 69, 275–295 (2016). https://doi.org/10.1007/s00025-016-0530-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-016-0530-2

Mathematics Subject Classification

Keywords

Search

Navigation