Open Access
2014 Lupaş-Durrmeyer operators based on Polya distribution
Vijay Gupta, Themistocles M. Rassias
Banach J. Math. Anal. 8(2): 146-155 (2014). DOI: 10.15352/bjma/1396640060
Abstract

The generalization of the Bernstein polynomials based on Polya distribution is considered in the present article. Here, we introduce a mixed summation-integral type operators having Polya and Bernstein basis functions in summation and integration respectively. We establish some direct results which include an asymptotic formula, local and global approximation results for these operators in terms of modulus of continuity.

References

1.

U. Abel, V. Gupta and R.N. Mohapatra, Local approximation by a variant of Bernstein–Durrmeyer operators, Nonlinear Anal. 68 (11) (2008), 3372–3381.  MR2401350 U. Abel, V. Gupta and R.N. Mohapatra, Local approximation by a variant of Bernstein–Durrmeyer operators, Nonlinear Anal. 68 (11) (2008), 3372–3381.  MR2401350

2.

P.N. Agrawal and V. Gupta, Simultaneous approximation by linear combination of modified Bernstein polynomials, Bull. Greek Math. Soc. 39 (1989), 29–43.  MR1109357 P.N. Agrawal and V. Gupta, Simultaneous approximation by linear combination of modified Bernstein polynomials, Bull. Greek Math. Soc. 39 (1989), 29–43.  MR1109357

3.

A. Aral, V. Gupta and R.P. Agarwal, Applications of $q$ Calculus in Operator Theory, Springer, New York, 2013.  MR3075547 A. Aral, V. Gupta and R.P. Agarwal, Applications of $q$ Calculus in Operator Theory, Springer, New York, 2013.  MR3075547

4.

L. Lupaş and A. Lupaş, Polynomials of binomial type and approximation operators, Studia Univ. Babes-Bolyai, Mathematica 32 no. 4 (1987), 61–69.  MR968181 L. Lupaş and A. Lupaş, Polynomials of binomial type and approximation operators, Studia Univ. Babes-Bolyai, Mathematica 32 no. 4 (1987), 61–69.  MR968181

5.

R.A. DeVore ang G.G. Lorentz, Constructive Approximation, Grundlehren der Mathematischen Wissenschaften, Band 303, Springer-Verlag, Berlin, Heidelberg, New York and London, 1993.  MR1261635 R.A. DeVore ang G.G. Lorentz, Constructive Approximation, Grundlehren der Mathematischen Wissenschaften, Band 303, Springer-Verlag, Berlin, Heidelberg, New York and London, 1993.  MR1261635

6.

Z. Ditzian and V. Totik, Moduli of smoothness, Springer Series in Computational Mathematics, 9. Springer-Verlag, New York, 1987.  MR914149 Z. Ditzian and V. Totik, Moduli of smoothness, Springer Series in Computational Mathematics, 9. Springer-Verlag, New York, 1987.  MR914149

7.

J.L. Durrmeyer, Une formule d' inversion de la Transformee Laplace, Applications a la Theorie des Moments, These de 3e Cycle, Faculte des Sciences de I' Universite de Paris, 1967. J.L. Durrmeyer, Une formule d' inversion de la Transformee Laplace, Applications a la Theorie des Moments, These de 3e Cycle, Faculte des Sciences de I' Universite de Paris, 1967.

8.

S.M. Farsani, On the boundedness and compactness of a certain integral operator , Banach J. Math. Anal. 7 no. 2 (2013), 86–102. MR3039941 S.M. Farsani, On the boundedness and compactness of a certain integral operator , Banach J. Math. Anal. 7 no. 2 (2013), 86–102. MR3039941

9.

V. Gupta and R.P. Agarwal, Convergence Estimates in Approximation Theory, Springer, New York, 2014. V. Gupta and R.P. Agarwal, Convergence Estimates in Approximation Theory, Springer, New York, 2014.

10.

V. Gupta and N. Ispir, On simultaneous approximation for some modified Bernstein-type operators, Int. J. Math. Math. Sci. 71 (2004), 3951–3958.  MR2129423 10.1155/S0161171204312317 V. Gupta and N. Ispir, On simultaneous approximation for some modified Bernstein-type operators, Int. J. Math. Math. Sci. 71 (2004), 3951–3958.  MR2129423 10.1155/S0161171204312317

11.

V. Gupta and P. Maheshwari, Bézier variant of a new Durrmeyer type operators, Riv. Mat. Univ. Parma (N.S.) 7 no. 2 (2003), 9–21.  MR2031837 V. Gupta and P. Maheshwari, Bézier variant of a new Durrmeyer type operators, Riv. Mat. Univ. Parma (N.S.) 7 no. 2 (2003), 9–21.  MR2031837

12.

D. Miclăuş, The revision of some results for Bernstein Stancu type operators, Carpathian J. Math. 28 (2) (2012), 289–300.  MR3027256 D. Miclăuş, The revision of some results for Bernstein Stancu type operators, Carpathian J. Math. 28 (2) (2012), 289–300.  MR3027256

13.

D.D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roum. Math. Pures Appl. 13 (1968), 1173–1194.  MR238001 D.D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roum. Math. Pures Appl. 13 (1968), 1173–1194.  MR238001

14.

G.V. Milovanović, D.S. Mitrinović and Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994.  MR1298187 G.V. Milovanović, D.S. Mitrinović and Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994.  MR1298187
Copyright © 2014 Tusi Mathematical Research Group
Vijay Gupta and Themistocles M. Rassias "Lupaş-Durrmeyer operators based on Polya distribution," Banach Journal of Mathematical Analysis 8(2), 146-155, (2014). https://doi.org/10.15352/bjma/1396640060
Published: 2014
Vol.8 • No. 2 • 2014
Back to Top