Abstract
The paper deals with a class of linear positive operators expressed by q-series. By using modulus of smoothness an upper bound of approximation error is determined. We identify functions for which theseoperators provide uniform approximation over noncompact intervals. A particular case is delivered.
Authors
Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Vijay Gupta
Schoole of Applied Sciences Netaji Subhas Institute of Technology New Delhi, India
Keywords
q-series, approximation process, modulus of smoothness, rate of convergence.
Paper coordinates
O. Agratini, V. Gupta, On the uniform convergence of a -series, Carpathian Journal of Mathematics, 32 (2016) no. 2, pp. 141-146.
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[1] Acar, T. and Ulusoy, G., Approximation by modified Sz´asz-Durrmeyer operators, Period. Math. Hungar., 72 (2016), No. 1, 64–75
[2] Agratini, O. and Radu, C., On q-Baskakov-Mastroianni operators, Rocky Mountain J. Math., 42 (2012), No. 3, 773–790
[3] Andrews, G. E., q-Series: Their Development and Applications in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, Conference Board of the Mathematical Sciences, Number 66, American Mathematical Society, 1986
[4] Aral, A., A generalization of Szasz-Mirakjan operators based on q-integers, Math. Comput. Modelling, 47 (2008), 1052–1062
[5] Aral, A. and Dogru, O., Bleimann, Butzer and Hahn operators on the q-integers, J. Ineq. Appl., Art. ID79410 (2007), 1–12
[6] Aral, A., Gupta, V., On certain q Baskakov-Durrmeyer operators, Creat. Math. Inform, 22 (2013), No. 1, 1–8
[7] Aral, A., Gupta, V. and Agarwal, R. P., Applications of q-Calculus in Operator Theory, Springer Science & Business Media New York, 2013
[8] Baskakov, V. A., An example of a sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk. SSSR, 113 (1957), 249–251
[9] Barbosu, D., Acu, A.- M. and Sofonea, F. D., The Voronovskaja-type formula for the Bleimann, Butzer and Hahn operators, Creat. Math. Inform., 23 (2014), No. 2, 137–140
[10] Barbosu, D., Pop, O. T. and Miclaus, D., On some extensions for the Szasz-Mirakjan operators, An. Univ. Oradea Fasc. Mat., 18 (2011), No. 1, 179–187
[11] Barbosu, D., Pop, O. T. and Miclaus, D., The Kantorovich form of some extensions for Szasz-Mirakjan operators Rev. Anal. Numer. Theor. Approx., 39 (2010), No. 1, 8–20
[12] Barbosu, D., Generalized blending operators of Favard-Szasz type, Bul. Stiint. Univ. Baia Mare Ser. B, 10 (1994), No. 1-2, 45–51
[13] Dogru, O. and Duman, O., Statistical approximation of Meyer-Konig and Zeller operators based on the q-integers, Publ. Math. Debrecen, 68 (2006), 199–214
[14] Gupta, V. and Sharma, H., Statistical approximation by q-integrated Meyer-Konig-Zeller-Kantorovich operators, Creat. Math. Inform., 19 (2010), No. 1, 45–52
[15] Lorentz, G. G., Bernstein Polynomials, 2nd Ed., Chelsea Publ. Comp., New York, NY, 1986
[16] Lupas, A., A q-analogue of the Bernstein operators, Seminar on Numerical and Statistical Calculus (Cluj-Napoca, 1987), 85–92, Preprint, 87-9, Univ. ”Babes-Bolyai”, Cluj-Napoca, 1987
[17] Kac, V. and Cheung, P., Quantum Calculus, Universitext, Springer-Verlag, New York, 2002
[18] Mahmudov, N. I., Approximation by the q-Szasz-Mirakjan operators, Abstr. Appl. Anal., 2012, Art. ID 754217, 16 pp.
[19] Mastroianni, G., Su un operatore lineare e positivo, Rend. Acc. Sc. Fis. Mat., Napoli, Serie IV, 46 (1979), 161–176
[20] Mishra, V. N. and Sharma, P., On approximation properties of Baskakov Schurer Szasz operators, Appl. Math. Comput., 281 (2016), 381–393
[21] Mursaleen, M., Alotaibi, A. and Ansari, K. J., On a Kantorovich Variant of (p; q)-Szasz-Mirakjan operators, J. Funct. Spaces 2016, Art. ID 1035253, 9 pp.
[22] Ostrovska, S., On the Lupa¸s q-analogue of the Bernstein operator, Rocky Mountain J. Math., 36 (2006), No. 5, 1615–1629
[23] Pop, O. T., Miclaus, D. and Barbosu, D., The Voronovskaja type theorem for a general class of Szasz-Mirakjan operators, Miskolc Math. Notes, 14 (2013), No. 1, 219–231
[24] Pop, O. T., Barbosu, D. and Miclaus, D., The Voronovskaja type theorem for an extension of Szasz-Mirakjan operators, Demonstratio Math., 45 (2012), No. 1, 107–115
[25] Trif, T., Meyer-Konig and Zeller operators based on the q-integers, Rev. Anal. Numer. Theor. Approx., 29 (2000), 221–229
[26] Sharma, H. and Gupta, C., On (p; q)-generalization of Szasz-Mirakyan Kantorovich operators, Boll. Unione Mat. Ital., 8 (2015), No. 3, 213–222