Research Article
BibTex RIS Cite

Bivariate Cheney-Sharma operators on simplex

Year 2018, Volume: 47 Issue: 4, 793 - 804, 01.08.2018

Abstract

In this paper, we consider bivariate Cheney-Sharma operators which are not the tensor product construction. Precisely, we show that these operators preserve Lipschitz condition of a given Lipschitz continuous function $f$ and also the properties of the modulus of continuity function
when $f$ is a modulus of continuity function.

References

  • Agratini, O. and Rus, I. A. Iterates of a class of dicsrete linear operators via contraction principle, Comment. Math. Univ. Carolin. 44(3), 555-563, 2003.
  • Altomare, F. and Campiti, M. Korovkin-type approximaton theory and its applications, (Walter de Gruyter, Berlin-New York, 1994).
  • Başcanbaz-Tunca, G. and Tuncer, Y. On a Chlodovsky variant of a multivariate beta operator, J. Comput. Appl. Math. 235(16), 4816-4824, 2011.
  • Başcanbaz-Tunca, G. and Taşdelen, F. On Chlodovsky form of the Meyer-König and Zeller operators, An. Univ. Vest Timiş. Ser. Mat.- Inform. 49(2), 137-144, 2011.
  • Başcanbaz-Tunca, G., ݝnce-ݝlarslan, H. G. and Erençin, A. Bivariate Bernstein type operators, Appl. Math. Comput. 273, 543-552, 2016.
  • Başcanbaz-Tunca, G., Erençin, A. and Taşdelen, F. Some properties of Bernstein type Cheney and Sharma operators(accepted).
  • Brown, B.M., Elliott, D. and Paget, D.F. Lipschitz constants for the Bernstein polynomials of a Lipschitz continuous function, J. Approx. Theory 49(2), 196-199, 1987.
  • Cao, F. Modulus of continuity, K-functional and Stancu operator on a simplex, Indian J. Pure Appl. Math. 35(12), 1343-1364, 2004.
  • Cao, F., Ding, C. and Xu, Z. On multivariate Baskakov operator, J. Math. Anal. Appl. 307(1), 274-291, 2005.
  • Catinaş, T. and Otrocol, D. Iterates of multivariate Cheney-Sharma operators, J. Comput. Anal. Appl. 15(7), 1240-1246, 2013.
  • Cheney, E. W. and Sharma, A. On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma 2(5), 77-84, 1964.
  • Craciun, M. Approximation operators constructed by means of Sheer sequences, Rev. Anal. Numér. Théor. Approx. 30(2), 135-150, 2001.
  • Ding, C. and Cao, F. K-functionals and multivariate Bernstein polynomials, J. Approx. Theory, 155(2), 125-135, 2008.
  • Erençin, A., Başcanbaz-Tunca, G. and Taşdelen, F. Some preservation properties of MKZ-Stancu type operators, Sarajevo J. Math., 10(22)(1), 93-102, 2014.
  • Erençin, A., Başcanbaz-Tunca, G. and Taşdelen, F. Some properties of the operators defined by Lupaş, Rev. Anal. Numér. Théor. Approx. 43(2), 168-174, 2014.
  • Farcaş, M. D. About Bernstein polynomials, An. Univ. Craiova Ser. Mat. Inform. 35, 117- 121, 2008.
  • Khan, M. K. Approximation properties of Beta operators, Progress in approximation theory, (Academic Press, Boston, MA, 1991), 483-495.
  • Khan, M. K. and Peters, M. A. Lipschitz constants for some approximation operators of a Lipschitz continuous function, J. Approx. Theory 59(3), 307-315, 1989.
  • Li, Z. Bernstein polynomials and modulus of continuity, J. Approx. Theory 102(1), 171-174, 2000.
  • Lindvall, T. Bernstein polynomials and the law of large numbers, Math. Sci. 7(2), 127-139, 1982.
  • Stancu, D. D. and Cismaşiu, C. On an approximating linear positive operator of Cheney- Sharma, Rev. Anal. Numér. Théor. Approx. 26(1-2), 221-227, 1997.
  • Stancu, D. D., Cabulea, L. A. and Pop, D. Approximation of bivariate functions by means of the operators $S_{m,n}^{\alpha,\beta;a,b}$, Stud. Univ. Babeş-Bolyai Math. 47(4), 105-113, 2002.
  • Stancu, D. D. Use of an identity of A. Hurwitz for construction of a linear positive operator of approximation, Rev. Anal. Numér. Théor. Approx. 31(1), 115-118, 2002.
  • Stancu, D. D. and Stoica, E. I. On the use Abel-Jensen type combinatorial formulas for construction and investigation of some algebraic polynomial operators of approximation, Stud. Univ. Babeş-Bolyai Math. 54(4), 167-182, 2009.
  • Stoica, E. I. On the combinatorial identities of Abel-Hurwitz type and their use in constructive theory of functions , Stud. Univ. Babeş-Bolyai Math. 55(4), 249-257, 2010.
  • Taşcu, I. Approximation of bivariate functions by operators of Stancu-Hurwitz type, Facta Univ. Ser. Math. Inform. 20, 33-39, 2005.
  • Taşcu, I. and Horvat-Marc, A. Construction of Stancu-Hurwitz operator for two variables, Acta Univ. Apulensis Math. Inform. 11, 97-101, 2006.
  • Trif, T. An elementary proof of the preservation of Lipschitz constants by the Meyer-König and Zeller operators, J. Inequal. Pure Appl. Math. 4(5), Article90, 3pp., 2003.
Year 2018, Volume: 47 Issue: 4, 793 - 804, 01.08.2018

Abstract

References

  • Agratini, O. and Rus, I. A. Iterates of a class of dicsrete linear operators via contraction principle, Comment. Math. Univ. Carolin. 44(3), 555-563, 2003.
  • Altomare, F. and Campiti, M. Korovkin-type approximaton theory and its applications, (Walter de Gruyter, Berlin-New York, 1994).
  • Başcanbaz-Tunca, G. and Tuncer, Y. On a Chlodovsky variant of a multivariate beta operator, J. Comput. Appl. Math. 235(16), 4816-4824, 2011.
  • Başcanbaz-Tunca, G. and Taşdelen, F. On Chlodovsky form of the Meyer-König and Zeller operators, An. Univ. Vest Timiş. Ser. Mat.- Inform. 49(2), 137-144, 2011.
  • Başcanbaz-Tunca, G., ݝnce-ݝlarslan, H. G. and Erençin, A. Bivariate Bernstein type operators, Appl. Math. Comput. 273, 543-552, 2016.
  • Başcanbaz-Tunca, G., Erençin, A. and Taşdelen, F. Some properties of Bernstein type Cheney and Sharma operators(accepted).
  • Brown, B.M., Elliott, D. and Paget, D.F. Lipschitz constants for the Bernstein polynomials of a Lipschitz continuous function, J. Approx. Theory 49(2), 196-199, 1987.
  • Cao, F. Modulus of continuity, K-functional and Stancu operator on a simplex, Indian J. Pure Appl. Math. 35(12), 1343-1364, 2004.
  • Cao, F., Ding, C. and Xu, Z. On multivariate Baskakov operator, J. Math. Anal. Appl. 307(1), 274-291, 2005.
  • Catinaş, T. and Otrocol, D. Iterates of multivariate Cheney-Sharma operators, J. Comput. Anal. Appl. 15(7), 1240-1246, 2013.
  • Cheney, E. W. and Sharma, A. On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma 2(5), 77-84, 1964.
  • Craciun, M. Approximation operators constructed by means of Sheer sequences, Rev. Anal. Numér. Théor. Approx. 30(2), 135-150, 2001.
  • Ding, C. and Cao, F. K-functionals and multivariate Bernstein polynomials, J. Approx. Theory, 155(2), 125-135, 2008.
  • Erençin, A., Başcanbaz-Tunca, G. and Taşdelen, F. Some preservation properties of MKZ-Stancu type operators, Sarajevo J. Math., 10(22)(1), 93-102, 2014.
  • Erençin, A., Başcanbaz-Tunca, G. and Taşdelen, F. Some properties of the operators defined by Lupaş, Rev. Anal. Numér. Théor. Approx. 43(2), 168-174, 2014.
  • Farcaş, M. D. About Bernstein polynomials, An. Univ. Craiova Ser. Mat. Inform. 35, 117- 121, 2008.
  • Khan, M. K. Approximation properties of Beta operators, Progress in approximation theory, (Academic Press, Boston, MA, 1991), 483-495.
  • Khan, M. K. and Peters, M. A. Lipschitz constants for some approximation operators of a Lipschitz continuous function, J. Approx. Theory 59(3), 307-315, 1989.
  • Li, Z. Bernstein polynomials and modulus of continuity, J. Approx. Theory 102(1), 171-174, 2000.
  • Lindvall, T. Bernstein polynomials and the law of large numbers, Math. Sci. 7(2), 127-139, 1982.
  • Stancu, D. D. and Cismaşiu, C. On an approximating linear positive operator of Cheney- Sharma, Rev. Anal. Numér. Théor. Approx. 26(1-2), 221-227, 1997.
  • Stancu, D. D., Cabulea, L. A. and Pop, D. Approximation of bivariate functions by means of the operators $S_{m,n}^{\alpha,\beta;a,b}$, Stud. Univ. Babeş-Bolyai Math. 47(4), 105-113, 2002.
  • Stancu, D. D. Use of an identity of A. Hurwitz for construction of a linear positive operator of approximation, Rev. Anal. Numér. Théor. Approx. 31(1), 115-118, 2002.
  • Stancu, D. D. and Stoica, E. I. On the use Abel-Jensen type combinatorial formulas for construction and investigation of some algebraic polynomial operators of approximation, Stud. Univ. Babeş-Bolyai Math. 54(4), 167-182, 2009.
  • Stoica, E. I. On the combinatorial identities of Abel-Hurwitz type and their use in constructive theory of functions , Stud. Univ. Babeş-Bolyai Math. 55(4), 249-257, 2010.
  • Taşcu, I. Approximation of bivariate functions by operators of Stancu-Hurwitz type, Facta Univ. Ser. Math. Inform. 20, 33-39, 2005.
  • Taşcu, I. and Horvat-Marc, A. Construction of Stancu-Hurwitz operator for two variables, Acta Univ. Apulensis Math. Inform. 11, 97-101, 2006.
  • Trif, T. An elementary proof of the preservation of Lipschitz constants by the Meyer-König and Zeller operators, J. Inequal. Pure Appl. Math. 4(5), Article90, 3pp., 2003.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Gülen Başcanbaz-tunca

Ayşegül Erençin

Hatice Gül İnce-ilarslan

Publication Date August 1, 2018
Published in Issue Year 2018 Volume: 47 Issue: 4

Cite

APA Başcanbaz-tunca, G., Erençin, A., & İnce-ilarslan, H. G. (2018). Bivariate Cheney-Sharma operators on simplex. Hacettepe Journal of Mathematics and Statistics, 47(4), 793-804.
AMA Başcanbaz-tunca G, Erençin A, İnce-ilarslan HG. Bivariate Cheney-Sharma operators on simplex. Hacettepe Journal of Mathematics and Statistics. August 2018;47(4):793-804.
Chicago Başcanbaz-tunca, Gülen, Ayşegül Erençin, and Hatice Gül İnce-ilarslan. “Bivariate Cheney-Sharma Operators on Simplex”. Hacettepe Journal of Mathematics and Statistics 47, no. 4 (August 2018): 793-804.
EndNote Başcanbaz-tunca G, Erençin A, İnce-ilarslan HG (August 1, 2018) Bivariate Cheney-Sharma operators on simplex. Hacettepe Journal of Mathematics and Statistics 47 4 793–804.
IEEE G. Başcanbaz-tunca, A. Erençin, and H. G. İnce-ilarslan, “Bivariate Cheney-Sharma operators on simplex”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 4, pp. 793–804, 2018.
ISNAD Başcanbaz-tunca, Gülen et al. “Bivariate Cheney-Sharma Operators on Simplex”. Hacettepe Journal of Mathematics and Statistics 47/4 (August 2018), 793-804.
JAMA Başcanbaz-tunca G, Erençin A, İnce-ilarslan HG. Bivariate Cheney-Sharma operators on simplex. Hacettepe Journal of Mathematics and Statistics. 2018;47:793–804.
MLA Başcanbaz-tunca, Gülen et al. “Bivariate Cheney-Sharma Operators on Simplex”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 4, 2018, pp. 793-04.
Vancouver Başcanbaz-tunca G, Erençin A, İnce-ilarslan HG. Bivariate Cheney-Sharma operators on simplex. Hacettepe Journal of Mathematics and Statistics. 2018;47(4):793-804.