[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.
[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
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.
[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.
[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,
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.