Weighted Approximation by Szasz-King type Operators

2 years ago

(original), Approximation Theory, Linear and positive approximation operators, not-at-ICTP, paper

Abstract

By using $q$ -calculus, in the present paper we construct Szasz type operators in King sense, this meaning the operators preserve the first and the third test function of Bohman-Korovkin theorem. Rate of local and global convergence is obtained in the frame of weighted spaces. The statistical approximation property of our operators is also revealed.

Authors

Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

Ogun Dogru
Gazi University, Ankara, Turkey

Keywords

q-integers; Positive linear operators; Statistical convergence; Weighted modulus of smoothness

Paper coordinates

O. Agratini, O. Dogru, Weighted Approximation by Szasz-King type Operators, Taiwanese Journal of Mathematics, 14 (2010) no. 4, pp. 1283-129,

PDF

https://ictp.acad.ro/wp-content/uploads/2023/03/2010d-Agratini-Dogru-Weighted-Approximation-by-Szasz-King-type-Operators-Taiwanese-Journal-of-Mathematics-14-4-1283-1296.pdf

About this paper

Journal

Taiwanese Journal of Mathematics

Publisher Name

The Mathematical Society of the Republic of China
Project Euclid??

DOI

Print ISSN

1027-5487

Online ISSN

2224-6851

google scholar link

1. M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Appplied Mathematics, Series 55, Issued June, 1964.
2. O. Agratini, Linear operators that preserve some test functions, Int. J. of Math. and Math. Sci., Art. ID 94136, (2006), 1-11.
3. A. Aral, A generalization of Szasz-Mirakyan operators based on q-integers, Mathematical and Computer Modelling, 47 (2008), 1052-1062.
4. A. Aral and O. Dogru, Bleimann, Butzer and Hahn operators based on the q-integers, J. Ineq. and Appl., Art. ID 79410, (2007), 1-12.
5. A. Aral and V. Gupta, The q-derivative and applications to q-Szasz Mirakyan operators, Calcolo, 43 (2006), 151-170.
6. Yuan-Chuan Li and Sen-Yen Shaw, A proof of Holder’s inequality using the Cauchy-Schwarz inequality, Journal of Inequalities in Pure and Applied Mathematics, 7 (2006), Issue 2, Article 62, 1-3.
7. M.-M. Derriennic, Modified Bernstein polynomials and Jacobi polynomials in qcalculus, Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl., 76 (2005), 269-290.
8. O. Dogru and O. Duman, Statistical approximation of Meyer-Konig and Zeller operators based on the q-integers, Publ. Math. Debrecen, 68 (2006), 199-214.
9. O. Duman and C. Orhan, Statistical approximation by positive linear operators, Studia Math., 161(2) (2004), 187-197.
10. H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.
11. J. A. Fridy and H. I. Miller, A matrix characterization of statistical convergence, Analysis, 11 (1991), 59-66.
12. A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32(1) (2002), 129-138.
13. G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge Univ. Press, 1990.
14. V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, 2002.
15. J. P. King, Positive linear operators which preserve x2, Acta Math. Hungar., 99(3) (2003), 203-208.
16. E. Kolk, Matrix summability of statistically convergent sequences, Analysis, (1993), 77-83.
17. A.-J. Lopez-Moreno, Weighted simultaneous approximation with Baskakov type operators, Acta Mathematica Hungarica, 104(1-2) (2004), 143-151.
18. A. Lupas, A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on Numerical and Statistical Calculus, 9 (1987), 85-92.
19. S. Ostrovska, q-Bernstein polynomials and their iterates, J. Approx. Th., 123 (2003), 232-255.
20. S. Ostrovska, On the Lupas¸ q-analogue of the Bernstein operator, Rocky Mountain Journal of Mathematics, 36(5) (2006), 1615-1629.
21. G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4 (1997), 511-518.
22. T. Trif, Meyer-Konig and Zeller operators based on the q-integers, Rev. Anal. Numer. Theor. Approx., 29 (2000), 221-229.
23. V. S. Videnskii, On some classes of q-parametric positive operators, Operator Theory Adv. Appl., 158 (2005), 213-222.
24. H. Wang, Korovkin-type theorem and application, J. Approx. Th., 132(2) (2005), 258-264.