Tham khảo luận văn - đề án 'báo cáo hóa học: "research article on the generalized favard-kantorovich and favard-durrmeyer operators in exponential function spaces"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 75142 12 pages doi 2007 75142 Research Article On the Generalized Favard-Kantorovich and Favard-Durrmeyer Operators in Exponential Function Spaces Grzegorz Nowak and Aneta Sikorska-Nowak Received 18 January 2007 Revised 12 June 2007 Accepted 14 November 2007 Recommended by Ulrich Abel We consider the Kantorovich- and the Durrmeyer-type modifications of the generalized Favard operators and we prove an inverse approximation theorem for functions f such that waf e Lp R where 1 p TO and wa x exp -ơx2 Ơ 0. Copyright 2007 G. Nowak and A. Sikorska-Nowak. 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. Preliminaries Let Lp ữ R f waf p TO for 1 p TO be the weighted function space where wa x exp -ơx2 Ơ 0 tc TO 1 P llgIIp I x pdx if 1 p TO - llg II TO essup g x . xeR We define the generalized Favard operators Fn for functions f R R by Fnf x X f k n pn k x y x e R n e N k -TO 2 Journal of Inequalities and Applications where N 1 2 . pn k x y 1_-expi--Wk - x ny J 2n 2y2n n and y yn TO 1 is a positive sequence convergent to zero see 1 . In the case where y n 9 2n with a positive constant 9 Fn become the known Favard operators introduced by Favard 2 . Some approximation properties of the classical Favard operators for continuous functions f on R are presented in 3 4 . Some approximation properties of their generalization can be found for example in 1 5 . Denote by F the Kantorovich-type modification of operators Fn defined by F f x n ị pnk x y f k 1 nf t dt x e R n e N k -TO jk n and by Fn the Durrmeyer-type modification of operators Fn Fnf x n ị pnJk x y r Pn k t y f t dt x e R n e N -to k -TO where f e Lp a R . Some estimates concerning the rates of pointwise convergence of the operators F f
