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Simultaneous Approximation Properties of q-Modified Beta Operators

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Mathematical Analysis and its Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 143))

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Abstract

We establish some approximation properties in simultaneous approximation for a q-analogue of the q-modified Beta operators \(B_n^q(f,x)\) introduced by Gupta and Kim. The convergence properties of the q-derivatives of these operators are discussed. Using the estimates for q-moments, the rate of approximation in simultaneous approximation is obtained in terms of modulus of continuity.

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References

  1. Aral, A., Gupta, V.: The \(q\)-derivative and applications to \(q\)-Szász Mirakyan operators. CALCOLO 43, 151–170 (2006)

    Google Scholar 

  2. Aral, A., Gupta, V., Agarwal, R.P.: Applications of \(q\)-Calculus in Operator Theory. Springer, New York (2013)

    Book  MATH  Google Scholar 

  3. De Sole, A., Kac, V.G.: On integral representations of \(q\)-gamma and \(q\)-beta functions. Rend. Mat. Acc. Lincei, s. 9. 16, 11–29 (2005)

    Google Scholar 

  4. DeVore, R.A, Lorentz, G.G.: Constructive Approximation, Springer-Verlag, Berlin, New York (1993)

    Google Scholar 

  5. Doğru, O., Duman, O.: Statistical approximation of Meyer-König and Zeller operators based on the \(q\)-integers. Publ. Math. Debrecen 68, 199–214 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Durrmeyer, J.L.: Une formula d’inversion, de la transformee de Laplace: Application a la theorie des Moments, These de 3e Cycle, Faculte des Sciences de l’universite de Paris (1967)

    Google Scholar 

  7. Ernst, T.: A new notation for \(q\)-calculus and a new \(q\)-Taylor formula, Uppsala University Report. Depart. Math. 1–28 (1999)

    Google Scholar 

  8. Gairola, A.R., Agrawal, P.N., Dobhal, G., Singh, K.K.: Moments of a \(q\)-Baskakov-beta operators in case 0 \(<\) q \(<\) 1. J. Classical Anal. 2(1), 9–22 (2013)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Gupta, V.: On certain \(q\)-Durrmeyer type operators. Appl. Math. Comput. 209, 415–420 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gupta, V., Heping, W.: The rate of convergence of \(q\)-Durrmeyer operators for 0 \(<\) q \(<\) 1. Math. Meth. Appl. Sci. 31, 1946–1955 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gupta, V., Kim, T.: On the rate of approximation by \(q\)-modified Beta operators. J. Math. Anal. Appl. 377, 471–480 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. II’inskii, A., Ostrovska, S.: Convergence of generalized Bernstein polynomials. J. Approx. Theor. 116(1), 100–112 (2002)

    Article  Google Scholar 

  14. Jackson, F.H.: On a \(q\)-definite integrals. Quarterly J. Pure Appl. Math. 41, 193–203 (1910)

    MATH  Google Scholar 

  15. Kac, V.G., Cheung, P.: Quantum Calculus. Universitext, Springer-Verlag, New York (2002)

    Book  MATH  Google Scholar 

  16. Kantorovich, L.V.: Sur certain developments suivant les polynomes de la forms de S. Bernstein, Dokl. Akad. Nauk. SSSR. 1–2, 563–568, 595–600 (1930)

    Google Scholar 

  17. Karsli, H., Gupta, V.: Some approximation properties of \(q\)-Chlodowsky operators. Appl. Math. Comput. 195, 220–229 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Koornwinder, T.H.: \(q\)-Special Functions, a tutorial. In: Gerstenhaber, M., Stasheff, J. (eds.) Deformation and Qauntum Groups with Applications of Mathematical Physics, Contemp. Math., 134 (1992), Amer. Math. Soc. (1992)

    Google Scholar 

  19. Mahmudov, N.: The moments for \(q\)-Bernstein operators in the case 0 \(<\) q \(<\) 1. Numer. Algor. 43–450 (2010)

    Google Scholar 

  20. Ostrovska, S.: \(q\)-Bernstein polynomials and their iterates. J. Approx. Theor. 123(2), 232–255 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Phillips, G.M.: Interpolation and Approximation by Polynomials, CMS Books in Mathematics, vol. 14. Springer, Berlin (2003)

    Book  Google Scholar 

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

    Google Scholar 

  24. Wang, H., Wu, X.: Saturation of convergence of \(q\)-Bernstein polynomials in the case q \(>\) 1. J. Math. Anal. Appl. 337(1), 744–750 (2008)

    Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are indebted to anonymous referees for many valuable comments especially in connection with Lemma 1 which helped to improve the paper.

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Authors and Affiliations

  1. University of Petroleum and Energy Studies, Dehradun, India

    Asha Ram Gairola & Girish Dobhal

  2. Government Polytechnic, Rampur, India

    Karunesh Kumar Singh

Authors
  1. Asha Ram Gairola
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  2. Girish Dobhal
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  3. Karunesh Kumar Singh
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Corresponding author

Correspondence to Asha Ram Gairola .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    P. N. Agrawal

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    R. N. Mohapatra

  3. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    Uaday Singh

  4. Dept. of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    H. M. Srivastava

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Gairola, A.R., Dobhal, G., Singh, K.K. (2015). Simultaneous Approximation Properties of q-Modified Beta Operators. In: Agrawal, P., Mohapatra, R., Singh, U., Srivastava, H. (eds) Mathematical Analysis and its Applications. Springer Proceedings in Mathematics & Statistics, vol 143. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2485-3_13

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