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Báo cáo hóa học: " Research Article Strong Converse Inequality for a Spherical Operator"

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 Strong Converse Inequality for a Spherical Operator | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 434175 7 pages doi 10.1155 2011 434175 Research Article Strong Converse Inequality for a Spherical Operator Shaobo Lin and Feilong Cao Institute of Metrology and Computational Science China Jiliang University Hangzhou 310018 Zhejiang Province China Correspondence should be addressed to Feilong Cao feilongcao@gmail.com Received 2 July 2010 Accepted 8 February 2011 Academic Editor S. S. Dragomir Copyright 2011 S. Lin and F. Cao. 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. In the paper titled as Jackson-type inequality on the sphere 2004 Ditzian introduced a spherical nonconvolution operator Otr which played an important role in the proof of the well-known Jackson inequality for spherical harmonics. In this paper we give the lower bound of approximation by this operator. Namely we prove that there are constants C1 and C2 such that C1M2r f t p Ot rf - f lip C2M2r f t p for any pth Lebesgue integrable or continuous function f defined on the sphere where w2r J t p is the 2rth modulus of smoothness of f. 1. Introduction Let sd-1 x e Rd x 1 d 3 be the unit sphere of Rd endowed with the usual rotation invariant measure do x . We denote by H-d the space of all spherical harmonics of degree d on sd-1 and nd the space of all spherical harmonics of degree at most n. The spaces H d 0 1 . are mutually orthogonal with respect to the inner product f g f x gCMiMM. d-1 1.1 so there holds nd Hd Hd --- H d. 1.2 2 Journal of Inequalities and Applications By C Sd-1 and Lp Sd-1 1 p TO we denote the space of continuous real-value functions and the space of the equivalence classes of p-integrable functions defined on sd-1 endowed with the respective norms f . 1 p llfllc Sd-1 max1 f p II lip f p pdcv p 1 p TO. L3 v 7 peSd-1 d-1 In the