Approximation with an arbitrary order by generalized Kantorovich-type and Durrmeyer-type operators on [0, +∞)
DOI:
https://doi.org/10.24193/subbmath.2017.4.07Keywords:
Generalized Sz´asz-Kantorovich operators, generalized Baskakov- Kantorovich operators, generalized Szssz-Durrmeyer-Stancu operators, generalized Baskakov-Szasz Durrmeyer-Stancu operators, linear and positive operators, modulus of continuity, arbitrary order of approximation.Abstract
Given an arbitrary sequence λn > 0, n ∈ N, with the property that limn→∞ λn = 0 as fast we want, in this note we introduce modified/ gene- ralized Sza´sz-Kantorovich, Baskakov-Kantorovich, Sza´sz-Durrmeyer-Stancu and Baskakov-Sza´sz-Durrmeyer-Stancu operators in such a way that on each compact subinterval in [0, +∞) the order of uniform approximation is ω1(f ; √λn). These modified operators uniformly approximate a Lipschitz 1 function, on each com- pact subinterval of [0, ∞) with the arbitrary good order of approximation √λn. The results obtained are of a definitive character (that is are the best possible) and also have a strong unifying character, in the sense that for various choices of the nodes λn, one can recapture previous approximation results obtained for these operators by other authors.
Mathematics Subject Classification (2010): 41A36, 41A25.
References
Agratini, O., Approximation by Linear Operators (Romanian), University Press, Babe¸s- Bolyai University, Cluj-Napoca, 2000.
Baskakov, V.A., An example of a sequence of linear positive operators in the space of continuous functions (Russian), Dokl. Akad. Nauk SSSR, 113(1957), 249-251.
Boehme, T.K., Bruckner, A.M., Functions with convex means, Pacific J. Math., 14(1964), 1137-1149.
Dieudonn´e, J., E´l´ements dAnalyse; 1. Fondements de l’Analyse Moderne, Gauthiers Villars, Paris, 1968.
Djebali, S., Uniform continuity and growth of real continuous functions, Int. J. Math. Education in Science and Technology, 32(2001), no. 5, 677-689.
Gal, S.G., Approximation with an arbitrary order by generalized Sz´asz-Mirakjan opera- tors, Stud. Univ. Babe¸s-Bolyai Math., 59(2014), no. 1, 77-81.
Gal S.G., Gupta, V., Approximation by complex Sz´asz-Durrmeyer operators in compact disks, Acta Math. Scientia, 34B(2014), no. 4, 1157-1165.
Gal, S.G., Gupta, V., Approximation by complex Sz´asz-Mirakjan-Stancu-Durrmeyer op- erators in compact disks under exponential growth, Filomat, 29(2015), no. 5, 1127-1136. [9] Gal, S.G., Opri¸s, D.B., Approximation with an arbitrary order by modified Baskakov type operators, Appl. Math. Comp., 265(2015), 329-332.
Gal, S.G., Opri¸s, D.B., Approximation of analytic functions with an arbitrary order by generalized Baskakov-Faber operators in compact sets, Complex Anal. Oper. Theory, 10(2016), no. 2, 369-377.
Gupta, V., Complex Baskakov-Sza´sz operators in compact semi-disks, Lobachevskii J. Math., 35(2014), no. 2, 65-73.
Gupta, V., Overconvergence of complex Baskakov-Sza´sz-Stancu operators, Mediterr. J. Math., 12(2015), no. 2, 455-470.
Mazhar, S.M., Totik V., Approximation by modified Sz´asz operators, Acta Sci. Math., 49(1985), 257-269.
Shisha, O., Mond, B., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 60(1968), 1196-1200.
Totik, V., Approximation by Sz´asz-Mirakjan-Kantorovich operators in Lp (p > 1) (Rus- sian), Analysis Math., 9(1983), no. 2, 147-167.
Walczak, Z., On approximation by modified Sz´asz-Mirakjan operators, Glasnik Mat., 37(2002), no. 2, 303-319.
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