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Báo cáo hóa học: "Research Article Exact Values of Bernstein n-Widths for Some Classes of Periodic Functions with Formal "

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 Exact Values of Bernstein n-Widths for Some Classes of Periodic Functions with Formal | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 894529 8 pages doi 10.1155 2008 894529 Research Article Exact Values of Bernstein n-Widths for Some Classes of Periodic Functions with Formal Self-Adjoint Linear Differential Operators Feng Guo Department of Mathematics Taizhou University Zhejiang Taizhou 317000 China Correspondence should be addressed to Feng Guo guofeng0576@163.com Received 10 December 2007 Accepted 18 June 2008 Recommended by Vijay Gupta We consider the classes of periodic functions with formal self-adjoint linear differential operators wp Lr which include the classical Sobolev class as its special case. With the help of the spectral of linear differential equations we find the exact values of Bernstein n-width of the classes wp Lr in the Lp for 1 p co. Copyright 2008 Feng Guo. 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. Introduction and main result Let C R Z N and N be the sets of all complex numbers real numbers integers nonnegative integers and positive integers respectively. Let T be the unit circle realized as the interval 0 2n with the points 0 and 2n identified and as usual let Lq Lq 0 2n be the classical Lebesgue integral space of 2n-periodic real-valued functions with the usual norm Hq 1 q OT. Denote by wp the Sobolev space of functions x - on T such that the r - 1 st derivative x r-fi is absolutely continuous on T and x Lp r G N. The corresponding Sobolev class is the set wp w l x r G llp 1 - 1.1 Tikhomirov 1 introduced the notion of Bernstein width of a centrally symmetric set C in a normed space X. It is defined by the following formula bn C X supsupỊẰ 0 L n XBX c C 1.2 L 2 Journal of Inequalities and Applications where BX is the unit ball of X and the outer supremum is taken over all subspaces L c X such that dim L n 1 n G