tailieunhanh - Báo cáo hóa học: " Research Article A New Estimate on the Rate of Convergence of ´ Durrmeyer-Bezier Operators"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A New Estimate on the Rate of Convergence of ´ Durrmeyer-Bezier Operators | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 702680 7 pages doi 2009 702680 Research Article A New Estimate on the Rate of Convergence of Durrmeyer-Bezier Operators Pinghua Wang1 and Yali Zhou2 1 Department of Mathematics Quanzhou Normal University Fujian 362000 China 2 Liming University Quanzhou Fujian 362000 China Correspondence should be addressed to Pinghua Wang xxc570@ Received 20 February 2009 Accepted 13 April 2009 Recommended by Vijay Gupta We obtain an estimate on the rate of convergence of Durrmeyer-Bezier operaters for functions of bounded variation by means of some probabilistic methods and inequality techniques. Our estimate improves the result of Zeng and Chen 2000 . Copyright 2009 P. Wang and Y. Zhou. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introdution In 2000 Zeng and Chen 1 introduced the Durrmeyer-Bézier operators Dn a which are defined as follows n p 1 Dn a f x n 2 Q x f Opnk t dt h1 k 0 J0 whom 6 is dpfinpd on ID 11 m 1 o a fvf ĩa ỉ__ ĩa fxl Ỉ-Í X n wnere f is uenneu on 1J a 1 Qnk x Jnk x Jn k 1 x Jnk x 2-1 j kpnj x k 0 1 2 . n are Bézier basis functions and pnk x n k n - k xk 1 -x n-k k 0 1 2 . n are Bernstein basis functions. When a 1 Dn 1 f is just the well-known Durrmeyer operator Dn 1 f x n 1 2 Pnk x f f t pnk t dt. k 0 J0 Concerning the approximation properties of operators Dn 1 f and some results on approximation of functions of bounded variation by positive linear operators one can refer 2 Journal of Inequalities and Applications to 2-7 . Authors of 1 studied the rate of convergence of the operators Dn a f for functions of bounded variation and presented the following important result. Theorem A. Let f be a function of bounded variation on 0 1 f e BV 0 1 a 1 then for every x e 0 1 and n 1 x 1 - x