Kantorovich-Stancu type operators including Boas-Buck type polynomials

Year 2019, Volume: 48 Issue: 2, 460 - 471, 01.04.2019

Bayram Çekim , Rabia Aktaş , Gürhan İçöz

Abstract

The aim of the paper is to introduce a Kantorovich-Stancu type modification of a generalization of Szász operators defined via Boas-Buck type polynomials and to obtain rates of convergence for these operators. Furthermore, we give the figures for comparing approximation properties of the operators $\mathcal{K}_{n}^{(\alpha,\beta)}$ and $\mathcal{B}_{n}.$

Keywords

Szász operator, modulus of continuity, rate of convergence, Boas-Buck type polynomials, Gould-Hopper polynomials, Charlier polynomials

References

• R. Aktaş, B. Çekim and F. Taşdelen, A Kantorovich-Stancu type generalization of Szász operators including Brenke-type polynomials, J. Func. Spaces 2013, 9 pages, 2013.
• A. Altın, O. Dogru and F. Tasdelen, The generalization of Meyer-Konig and Zeller operators by generating functions, J. Math. Anal. Appl. 312 (1), 181-194, 2005.
• F. Altomare and M. Campiti, Korovkin-type approximation theory and its applica- tions. Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff, de Gruyter Studies in Mathematics, 17. Walter de Gruyter & Co., Berlin, 1994.
• P.L. Butzer, On the extensions of Bernstein polynomials to the infinite interval, Proc. Amer. Soc. 5, 547-553, 1954.
• T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
• A. Ciupa, A class of integral Favard-Szász type operators, Stud. Univ. Babeş-Bolyai Math. 40 (1), 39-47, 1995.
• Z. Ditzian and V. Totik, Moduli of Smoothness, Springer-Verlag, New York, 1987.
• O. Dogru, M.A. Özarslan and F. Tasdelen, On positive operators involving a certain class of generating functions, Stud. Sci. Math. Hungar. 41 (4), 415-429, 2004.
• O. Duman, M.A. Özarslan and B. Della Vecchia, Modified q-Szász-Mirakjan- Kantorovich operators preserving linear functions, Turk. J. Math. 33 (2), 151-158, 2009.
• H.W. Gould and A.T. Hopper, Operational formulas connected with two generaliza- tions of Hermite polynomials, Duke Math. J. 29, 51-63, 1962.
• M.E.H. Ismail, On a generalization of Szász operators, Mathematica (Cluj), 39, 259- 267, 1974.
• M.E.H. Ismail, Classical and quantum orthogonal polynomial in one variable, Cam- bridge University Press, Cambridge, UK, 2005.
• G. İçöz, F. Taşdelen Yeşildal and O. Doğru, Kantrorovich Process of Linear Positive Operators Via Biorthogonal Polynomials, J. Inequal. Spec. Funct. 3, 77-84, 2012.
• G. İçöz, F. Taşdelen and S. Varma, On Linear Positive Operators Involving Biorthog- onal Polynomial, Ars Combin. 105, 319-331, 2012.
• G. İçöz, S. Varma and S. Sucu, Approximation by operators including generalized Appell polynomials, Filomat 30 (2), 429-440, 2016.
• A. Jakimovski and D. Leviatan, Generalized Szász operators for the approximation in the infinite interval, Mathematica (Cluj) 11, 97-103, 1969.
• A. Kajla and P. N. Agrawal, Szász-Kantorovich type operators based on Charlier polynomials, Kyungpook Math. J. 56 (3), 877-897, 2016.
• G. Nowak and A. Sikorska-Nowak, Some approximation properties for modified Szász- Mirakyan-Kantorovich operators, Rev. Anal. Numér. Théor. Approx. 38 (1), 73-82, 2009.
• M.A. Özarslan, q-Laguerre type linear positive operators, Stud. Sci. Math. Hungar. 44 (1), 65-80, 2007.
• M.A. Özarslan and O. Duman, Approximation properties of Poisson integrals for orthogonal expansions, Taiwanese J. Math. 12 (5), 1147-1163, 2008.
• S. Sucu, G. İçöz and S. Varma, On Some Extensions of Szász Operators Including Boas-Buck-Type Polynomials, Abstr. Appl. Anal. 2012, 15 pages, 2012.
• S Sucu and S Varma, Generalization of Jakimovski- Leviatan type Szász operators, Appl. Math. Comput. 270, 977-983, 2015.
• O. Szász, Generalization of S. Bernstein’s polynomials to n the infinite interval, J. Res. Natl. Bur. Stand. 45, 239-245, 1950.
• F. Taşdelen, R. Aktaş and A. Altın, A Kantorovich Type of Szász Operators Including Brenke-Type Polynomials, Abst. Appl. Anal. 2012, 13 pages, 2012.
• F. Taşdelen and A. Erençin, The generalization of bivariate MKZ operators by mul- tiple generating functions , J. Math. Anal. Appl. 331 (1), 727-735, 2007.
• V. Totik, Approximation by Szász-Mirakjan-Kantorovich operators in $L^{p}\left( p>1\right)$, Anal. Math. 9 (2), 147-167, 1983.
• S. Varma, S. Sucu and G. İçöz, Generalization of Szász operators involving Brenke type polynomials, Comput. Math. Appl. 64 (2), 121-127, 2012.
• S. Varma and F. Taşdelen, Szász type operators involving Charlier polynomials, Math- ematical and Computer Modelling 56 (5-6), 118-122, 2012.
• S. Varma and F. Taşdelen, On a generalization of Szász-Durrmeyer operators with some orthogonal polynomials, Stud. Univ. Babeş-Bolyai Math. 58 (2), 225-232, 2013.
• Z. Walczak, On approximation by modified Szász-Mirakyan operators, Glasnik Matematicki 37 (57), 303-319, 2002.