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Year 2020, , 510 - 522, 02.04.2020
https://doi.org/10.15672/hujms.458188

Abstract

References

  • [1] A.M. Acu, C.V.Muraru, D.F. Sofonea and A.V. Radu, Some approximation properties of a Durrmeyer variant of q-Bernstein–Schurer operators, Math. Methods Appl. Sci. 39 (18), 5636–5650, 2016.
  • [2] O. Agratini and O. Doğru, Weighted Approximation by q−szász King type operators, Taiwanese J. Math. 14 (4), 1283–1296, 2010.
  • [3] O. Agratini and I.A. Rus, Iterates of a class of dicsrete linear operators via contraction principle, Comment. Math. Univ. Carolin. 44 (3), 555–563, 2003.
  • [4] P.N. Agrawal, V. Gupta, A.S. Kumar and A. Kajla. Generalized Baskakov-Szász type operators, Appl. Math. Comput. 236, 311–324, 2014.
  • [5] F. Altomare and M. Campiti, Korovkin-type approximaton theory and its applications, Walter de Gruyter, Berlin-New York, 1994.
  • [6] G. Başcanbaz-Tunca and F. Taşdelen, On Chlodovsky form of the Meyer-König and Zeller operators, An. Univ. Vest Timiş. Ser. Mat.-Inform. 49 (2), 137–144, 2011.
  • [7] G. Başcanbaz-Tunca, A. Erençin and F. Taşdelen, Some properties of Bernstein type Cheney and Sharma operators, Gen. Math. 24 (1-2), 17–25, 2016.
  • [8] G. Başcanbaz-Tunca, A. Erençin and H. G. İnce-İlarslan, Bivariate Cheney and Sharma Operators on Simplex, Hacet. J. Math. Stat. 47 (4), 793–804, 2018.
  • [9] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford, 2000.
  • [10] R.C. Buck, The measure theoretic approach to density, Amer. J. Math. 68, 560-580, 1946.
  • [11] R.C. Buck, Generalized asymptotic density, Amer. J. Math. 75, 335–346, 1953.
  • [12] I. Büyükyazıcı, Approximation by Stancu-Chlodowsky polynomials, Comput. Math. Appl. 59, 274–282, 2010.
  • [13] E.W. Cheney and A. Sharma, On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma 2 (5), 77–84, 1964.
  • [14] I. Chlodovsky, Sur le development des fonctions dUfines dans un interval infinien series de polyn omes de S.N. Bernstein, Compositio Math. 4, 380–392, 1937.
  • [15] M. Craciun, Approximation operators constructed by means of Sheffer sequences, Rev. Anal. Numér. Théor. Approx. 30 (2), 135–150, 2001.
  • [16] O. Duman, M.K. Khan, and C. Orhan, A-statistical convergence of approximating operators, Math. Inequal. Appl. 6 (4), 689–699, 2003.
  • [17] O. Duman and C. Orhan, Statistical approximation by positive linear operators, Studia Math. 161 (2), 187–197, 2004.
  • [18] H. Fast, Sur la convergence statistique, Colloq. Math. 2, 241–244, 1951.
  • [19] J.A. Fridy, On statistical convergence, Analysis, 5 (4), 301–313, 1985.
  • [20] J.A. Fridy and H.I. Miller, A matrix characterization of statistical convergence, Anal- ysis 11, 59–66, 1991.
  • [21] A.D. Gadjiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogues to that of P.P. Korovkin, Dokl. Akad. Nauk SSSR 218, 1974.
  • [22] A.D. Gadjiev, Theorems of the type of P. P. Korovkin’s theorems (in Russian), Math. Z. 205, 781–786, 1976. Translated in Maths Notes, 20 (5-6), 995–998, 1977.
  • [23] A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32, 129–138, 2002.
  • [24] A.D. Gadjiev, R.O. Efendiev and E. Ibikli, Generalized Bernstein Chlodowsky poly- nomials, Rocky Mountain J. Math. 28 (4), 1267–1277, 1998.
  • [25] A. Izgi, Rate of approximation new-type generalization of Chlodovsky polynomial, Gen. Math. 21 (2), 125–140, 2013.
  • [26] H. Karsli and E. Ibikli, Convergence rate of a new Bezier variant of Chlodowsky operators to bounded variation functions, J. Comput. Appl. Math. 212 (2), 431–443, 2008.
  • [27] H. Karsli and V. Gupta, Some approximation properties of q−Chlodowsky operators, Appl. Math. Comput. 195 (1), 220–229, 2008.
  • [28] A.S. Kumar and T. Acar, Approximation by generalized Baskakov–Durrmeyer–Stancu type operators, Rend. Circ. Mat. Palermo, 65 (3), 411–424, 2016.
  • [29] M. Mursaleen, K.J. Ansari and A. Khan, Some approximation results by (p, q)−analogue of Bernstein–Stancu operators, Appl. Math. Comput. 264, 392–402, 2015.
  • [30] M. Mursaleen, A.H. Al-Abieda and A.M. Acu, Approximation by Chlodowsky type of Szász operators based on Boas-Buck type polynomials, Turk. J. Math 42, 2243–2259, 2018.
  • [31] M.A. Özarslan, O. Duman, and H.M. Srivastava, Statistical approximation results for Kantorovich-type operators involving some special polynomials, Math. Comput. Model. 48 (3-4), 388–401, 2008.
  • [32] D. Söylemez and M. Unver, Korovkin Type Theorems for Cheney-Sharma Operators via Summability Methods, Results Math. 72, 1601–1612, 2017.
  • [33] M.W. Müller, Approximation by Cheney-Sharma-Kantorovic Polynomials in the Lp- Metric, Rocky Mountain J. Math. 19 (1), 281–291, 1989.
  • [34] M. Unver, Abel transform of positive linear operators, in: ICNAAM 2013, AIP Con- ference Proceedings 1558, 1148–1151, 2013.
  • [35] D.D. Stancu and C. Cismaşiu, On an approximating linear positive operator of Cheney-Sharma, Rev. Anal. Numèr. Thèor. Approx. 26 (1-2), 221–227, 1997.
  • [36] D.D. Stancu and E.I. Stoica, On the use Abel-Jensen type combinatorial formulas for construction and investigation of some algebraic polynomial operators of approxima- tion, Stud. Univ. Babeş-Bolyai Math. 54 (4), 167–182,

Approximation by Cheney-Sharma Chlodovsky operators

Year 2020, , 510 - 522, 02.04.2020
https://doi.org/10.15672/hujms.458188

Abstract

The main purpose of this paper is to construct Cheney-Sharma Chlodovsky operators. We study approximation properties of the new operators with the help of weighted Korovkin-type theorem and universal Korovkin-type theorem. We also give the rate of convergence by means of the modulus of continuity. Furthermore, we give $A$-statistical convergence property of these operators.

References

  • [1] A.M. Acu, C.V.Muraru, D.F. Sofonea and A.V. Radu, Some approximation properties of a Durrmeyer variant of q-Bernstein–Schurer operators, Math. Methods Appl. Sci. 39 (18), 5636–5650, 2016.
  • [2] O. Agratini and O. Doğru, Weighted Approximation by q−szász King type operators, Taiwanese J. Math. 14 (4), 1283–1296, 2010.
  • [3] O. Agratini and I.A. Rus, Iterates of a class of dicsrete linear operators via contraction principle, Comment. Math. Univ. Carolin. 44 (3), 555–563, 2003.
  • [4] P.N. Agrawal, V. Gupta, A.S. Kumar and A. Kajla. Generalized Baskakov-Szász type operators, Appl. Math. Comput. 236, 311–324, 2014.
  • [5] F. Altomare and M. Campiti, Korovkin-type approximaton theory and its applications, Walter de Gruyter, Berlin-New York, 1994.
  • [6] G. Başcanbaz-Tunca and F. Taşdelen, On Chlodovsky form of the Meyer-König and Zeller operators, An. Univ. Vest Timiş. Ser. Mat.-Inform. 49 (2), 137–144, 2011.
  • [7] G. Başcanbaz-Tunca, A. Erençin and F. Taşdelen, Some properties of Bernstein type Cheney and Sharma operators, Gen. Math. 24 (1-2), 17–25, 2016.
  • [8] G. Başcanbaz-Tunca, A. Erençin and H. G. İnce-İlarslan, Bivariate Cheney and Sharma Operators on Simplex, Hacet. J. Math. Stat. 47 (4), 793–804, 2018.
  • [9] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford, 2000.
  • [10] R.C. Buck, The measure theoretic approach to density, Amer. J. Math. 68, 560-580, 1946.
  • [11] R.C. Buck, Generalized asymptotic density, Amer. J. Math. 75, 335–346, 1953.
  • [12] I. Büyükyazıcı, Approximation by Stancu-Chlodowsky polynomials, Comput. Math. Appl. 59, 274–282, 2010.
  • [13] E.W. Cheney and A. Sharma, On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma 2 (5), 77–84, 1964.
  • [14] I. Chlodovsky, Sur le development des fonctions dUfines dans un interval infinien series de polyn omes de S.N. Bernstein, Compositio Math. 4, 380–392, 1937.
  • [15] M. Craciun, Approximation operators constructed by means of Sheffer sequences, Rev. Anal. Numér. Théor. Approx. 30 (2), 135–150, 2001.
  • [16] O. Duman, M.K. Khan, and C. Orhan, A-statistical convergence of approximating operators, Math. Inequal. Appl. 6 (4), 689–699, 2003.
  • [17] O. Duman and C. Orhan, Statistical approximation by positive linear operators, Studia Math. 161 (2), 187–197, 2004.
  • [18] H. Fast, Sur la convergence statistique, Colloq. Math. 2, 241–244, 1951.
  • [19] J.A. Fridy, On statistical convergence, Analysis, 5 (4), 301–313, 1985.
  • [20] J.A. Fridy and H.I. Miller, A matrix characterization of statistical convergence, Anal- ysis 11, 59–66, 1991.
  • [21] A.D. Gadjiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogues to that of P.P. Korovkin, Dokl. Akad. Nauk SSSR 218, 1974.
  • [22] A.D. Gadjiev, Theorems of the type of P. P. Korovkin’s theorems (in Russian), Math. Z. 205, 781–786, 1976. Translated in Maths Notes, 20 (5-6), 995–998, 1977.
  • [23] A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32, 129–138, 2002.
  • [24] A.D. Gadjiev, R.O. Efendiev and E. Ibikli, Generalized Bernstein Chlodowsky poly- nomials, Rocky Mountain J. Math. 28 (4), 1267–1277, 1998.
  • [25] A. Izgi, Rate of approximation new-type generalization of Chlodovsky polynomial, Gen. Math. 21 (2), 125–140, 2013.
  • [26] H. Karsli and E. Ibikli, Convergence rate of a new Bezier variant of Chlodowsky operators to bounded variation functions, J. Comput. Appl. Math. 212 (2), 431–443, 2008.
  • [27] H. Karsli and V. Gupta, Some approximation properties of q−Chlodowsky operators, Appl. Math. Comput. 195 (1), 220–229, 2008.
  • [28] A.S. Kumar and T. Acar, Approximation by generalized Baskakov–Durrmeyer–Stancu type operators, Rend. Circ. Mat. Palermo, 65 (3), 411–424, 2016.
  • [29] M. Mursaleen, K.J. Ansari and A. Khan, Some approximation results by (p, q)−analogue of Bernstein–Stancu operators, Appl. Math. Comput. 264, 392–402, 2015.
  • [30] M. Mursaleen, A.H. Al-Abieda and A.M. Acu, Approximation by Chlodowsky type of Szász operators based on Boas-Buck type polynomials, Turk. J. Math 42, 2243–2259, 2018.
  • [31] M.A. Özarslan, O. Duman, and H.M. Srivastava, Statistical approximation results for Kantorovich-type operators involving some special polynomials, Math. Comput. Model. 48 (3-4), 388–401, 2008.
  • [32] D. Söylemez and M. Unver, Korovkin Type Theorems for Cheney-Sharma Operators via Summability Methods, Results Math. 72, 1601–1612, 2017.
  • [33] M.W. Müller, Approximation by Cheney-Sharma-Kantorovic Polynomials in the Lp- Metric, Rocky Mountain J. Math. 19 (1), 281–291, 1989.
  • [34] M. Unver, Abel transform of positive linear operators, in: ICNAAM 2013, AIP Con- ference Proceedings 1558, 1148–1151, 2013.
  • [35] D.D. Stancu and C. Cismaşiu, On an approximating linear positive operator of Cheney-Sharma, Rev. Anal. Numèr. Thèor. Approx. 26 (1-2), 221–227, 1997.
  • [36] D.D. Stancu and E.I. Stoica, On the use Abel-Jensen type combinatorial formulas for construction and investigation of some algebraic polynomial operators of approxima- tion, Stud. Univ. Babeş-Bolyai Math. 54 (4), 167–182,
There are 36 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Dilek Söylemez 0000-0002-6802-8064

Fatma Taşdelen Yeşildal 0000-0002-6291-1649

Publication Date April 2, 2020
Published in Issue Year 2020

Cite

APA Söylemez, D., & Taşdelen Yeşildal, F. (2020). Approximation by Cheney-Sharma Chlodovsky operators. Hacettepe Journal of Mathematics and Statistics, 49(2), 510-522. https://doi.org/10.15672/hujms.458188
AMA Söylemez D, Taşdelen Yeşildal F. Approximation by Cheney-Sharma Chlodovsky operators. Hacettepe Journal of Mathematics and Statistics. April 2020;49(2):510-522. doi:10.15672/hujms.458188
Chicago Söylemez, Dilek, and Fatma Taşdelen Yeşildal. “Approximation by Cheney-Sharma Chlodovsky Operators”. Hacettepe Journal of Mathematics and Statistics 49, no. 2 (April 2020): 510-22. https://doi.org/10.15672/hujms.458188.
EndNote Söylemez D, Taşdelen Yeşildal F (April 1, 2020) Approximation by Cheney-Sharma Chlodovsky operators. Hacettepe Journal of Mathematics and Statistics 49 2 510–522.
IEEE D. Söylemez and F. Taşdelen Yeşildal, “Approximation by Cheney-Sharma Chlodovsky operators”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 2, pp. 510–522, 2020, doi: 10.15672/hujms.458188.
ISNAD Söylemez, Dilek - Taşdelen Yeşildal, Fatma. “Approximation by Cheney-Sharma Chlodovsky Operators”. Hacettepe Journal of Mathematics and Statistics 49/2 (April 2020), 510-522. https://doi.org/10.15672/hujms.458188.
JAMA Söylemez D, Taşdelen Yeşildal F. Approximation by Cheney-Sharma Chlodovsky operators. Hacettepe Journal of Mathematics and Statistics. 2020;49:510–522.
MLA Söylemez, Dilek and Fatma Taşdelen Yeşildal. “Approximation by Cheney-Sharma Chlodovsky Operators”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 2, 2020, pp. 510-22, doi:10.15672/hujms.458188.
Vancouver Söylemez D, Taşdelen Yeşildal F. Approximation by Cheney-Sharma Chlodovsky operators. Hacettepe Journal of Mathematics and Statistics. 2020;49(2):510-22.