Rates of weighted statistical convergence for a generalization of positive linear operators

Reyhan Canatan Ilbey; Ogün Doğru

doi:10.3934/mfc.2022059

Article Contents

2023, Volume 6, Issue 3: 427-438. Doi: 10.3934/mfc.2022059

This issue Previous Article On Szász-Mirakyan-Jain Operators preserving exponential functions Next Article On wavelet type generalized Bézier operators

Rates of weighted statistical convergence for a generalization of positive linear operators

Reyhan Canatan Ilbey^1, , and
Ogün Doğru^2,

1.
Baskent University, Kahramankazan Vocational School, Kahramankazan 06980, Ankara, Turkey
2.
Gazi University, Faculty of Science, Department of Mathematics, Teknik Okullar 06500, Ankara, Turkey

^*Corresponding author: Reyhan Canatan Ilbey
^*Corresponding author: Reyhan Canatan Ilbey

Dedicated to Professor Vijay Gupta on the occasion of his 60th birthday

Received: October 2022

Revised: November 2022

Early access: December 26, 2022

Published online: December 26, 2022

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In the present paper, some direct and inverse theorems relating to a generalization of positive linear operators are given. Also some rates of weighted statistical convergence are computed by means of a weighted modulus of continiuty.

Keywords:

Mathematics Subject Classification: 41A10, 41A25, 41A36.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	N. I. Achieser, Vorlesungen über Approximationstheorie, Akademic-Verlag, Berlin, 1967.
[2]	O. Agratini, Korovkin type error estimates for Meyer-König and Zeller operators, Math. Ineq. Appl., 4 (2001), 119-126. doi: 10.7153/mia-04-09.
[3]	O. Agratini and O. Doğru, Weighted approximation by $q-$ Szász-King type operators, Taiwanese J. Math., 14 (2010), 1283-1296. doi: 10.11650/twjm/1500405945.
[4]	P. N. Agrawal, H. Karslı and M. Goyal, Szász-Baskakov type operators based on $q-$integers, J. Inequal. Appl., 441 (2014), 18 pp. doi: 10.1186/1029-242X-2014-441.
[5]	F. Altomore and M. Campiti, Korovkin Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics, 17. Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110884586.
[6]	N. T. Amanov, On the weighted approximation by Szász-Mirakyan operators, Anal. Math., 18 (1992), 167-184. doi: 10.1007/BF01911084.
[7]	V. A. Baskakov, An example of a sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR, 113 (1957), 249-251.
[8]	M. Becker, Global approximation theorems for Szász-Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. Journal, 27 (1978), 127-142. doi: 10.1512/iumj.1978.27.27068.
[9]	M. Becker, An elementary proof of the inverse theorem for Bernstein polynomials, Aequat. Math., 17 (1978), 389-390. doi: 10.1007/BF01818583.
[10]	H. Berens and G. G. Lorentz, Inverse theorems for Bernstein polynomials, Indiana University Math. Journal, 21 (1972), 693-708. doi: 10.1512/iumj.1972.21.21054.
[11]	G. Bleimann, P. L. Butzer and L. Hahn, A Bernstein-type operator approximating continuous functions on the semi-axis, Nederl. Akad. Wetensch. Indag. Math., 42 (1980), 255-262.
[12]	J. Bustamante, Approximation of functions and Mihesan operators, Mathematical Foundations of Computing, (2022), 1-10. doi: 10.3934/mfc.2022033.
[13]	P. L. Butzer and H. Berens, Semi-Groups of Operators and Approximation, Die Grundlehren der Mathematischen Wissenschaften, Band 145 Springer-Verlag New York, Inc., New York, 1967. doi: 10.1007/978-3-642-64981-3.
[14]	R. Canatan and O. Doğru, Statistical approximation properties of a generalization of positive linear operators, European J. of Pure and Applied Math., 5 (2012), 75-87.
[15]	E. W. Cheney and A. Sharma, Bernstein power series, Canadian J. Math., 16 (1964), 241-252. doi: 10.4153/CJM-1964-023-1.
[16]	E. Deniz, Quantitative estimates for Jain-Kantorovich operators, Commun. Fac. Sci. Univ. Ank. Ser. A1, Math. and Stat., 65 (2016), 121-132. doi: 10.1501/Commua1_0000000764.
[17]	R. A. DeVore, The Approximation of Continuous Functions by Positive Linear Operators, Lecture Notes in Mathematics, Vol. 293. Springer-Verlag, Berlin-New York, 1972.
[18]	R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren der mathematischen Wissenschaften, 303. Springer-Verlag, Berlin, 1993.
[19]	O. Doğru, Approximation order asymptotic approximation for generalized Meyer-König and Zeller operators, Math. Balkanika, 12 (1998), 359-367.
[20]	O. Doğru, Weighted approximation of continuous functions on the all positive axis by modifed linear positive operators, Int. J. Comput. Numer. Anal. Appl., 1 (2002), 135-147.
[21]	O. Doğru, Direct and inverse theorems on statistical approximations by positive linear operators, Rocky Mountain J. Math., 38 (2008), 1959-1973. doi: 10.1216/RMJ-2008-38-6-1959.
[22]	O. Doğru, Approximation properties of a generalization of positive linear operators, J. Math. Anal. Appl., 342 (2008), 161-170. doi: 10.1016/j.jmaa.2007.12.007.
[23]	O. Doğru, Weighted approximation properties of Szász-type operators, Int. Math. Journal, 2 (2002), 889-895.
[24]	H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244. doi: 10.4064/cm-2-3-4-241-244.
[25]	G. Freud, Investigations on weighted approximation by polynomials, Studia Sci. Math. Hungar., 8 (1973), 285-305.
[26]	J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301-313. doi: 10.1524/anly.1985.5.4.301.
[27]	A. D. Gadjiev, The convergence problem for a sequences of positive lnear operators on unbounded sets, and theorems analogous to that of P. P. Korovkin, Soviet Math. Dokl., 15 (1974), 1433-1436.
[28]	A. D. Gadjiev, On Korovkin type theorems, Math. Zametki, 20 (1976), 781-786.
[29]	A. D. Gadjiev and A. Aral, The estimates of approximation by using a new type of weighted modulus of continuity, Computers and Mathematics with Applications, 54 (2007), 127-135. doi: 10.1016/j.camwa.2007.01.017.
[30]	E. A. Gadjieva and O. Doğru, Weighted approximation properties of Szász operators to continuous functions. Ⅱ, Kizilirmak Int. Sci. Conference Proc., Kirikkale Univ. Turkey, (1998), 29-37.
[31]	A. Holhoş, Quantitative estimates for positive linear operators in weighted spaces, General Mathematics, 16 (2008), 99-110.
[32]	A. Holhoş, Uniform weighted approximation by positive linear operators, Stud. Univ. Babeş Bolyai Math., 56 (2011), 135-146.
[33]	N. Ispir, On modifed Baskakov operators on weighted spaces, Turk. J. Math., 26 (2001), 355-365.
[34]	A.-J. Lopez-Moreno, Weighted simultaneous approximation with Baskakov type operators, Acta Math. Hungar., 104 (2004), 143-151. doi: 10.1023/B:AMHU.0000034368.81211.23.
[35]	G. G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, Inc., Athena Series, New York, 1966.
[36]	V. Mihesan, Uniform approximation with positive linear operators generated by generalized Baskakov method, Autmat. Comput. Appl. Math., 7 (1998), 34-37.
[37]	G. M. Mirakjan, Approximation of continuous functions with the aid of polynomials, Dokl. Akad. Nauk SSSR, 31 (1941), 201-205.
[38]	O. Szász, Generalization of S. Bernstein's polynomials to the infinite interval, J. Research Nat. Bur. Standards, 45 (1950), 239-245. doi: 10.6028/jres.045.024.