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 The Direct and Converse Inequalities for Jackson-Type Operators on Spherical Cap | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 205298 16 pages doi 2009 205298 Research Article The Direct and Converse Inequalities for Jackson-Type Operators on Spherical Cap Yuguang Wang and Feilong Cao Institute of Metrology and Computational Sciences China Jiliang University Hangzhou 310018 China Correspondence should be addressed to Feilong Cao feilongcao@ Received 10 July 2009 Accepted 23 October 2009 Recommended by Marta A D Garcia-Huidobro Approximation on the spherical cap is different from that on the sphere which requires us to construct new operators. This paper discusses the approximation on the spherical cap. That is the so-called Jackson-type operator fms _1 is constructed to approximate the function defined on the spherical cap D x0 Y . We thus establish the direct and inverse inequalities and obtain saturation theorems for fms _1 on the cap D x0 y . Using methods of K-functional and multiplier we obtain the inequality C1WJfs f - f d p w2 f k 1 D p C2maXv k Jfs f - f d p and that the saturation order of these operators is O k-2 where u3 f 1 dp is the modulus of smoothness of degree 2 the constants C1 and C2 are independent of k and f. Copyright 2009 Y. Wang 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. 1. Introduction In the past decades many mathematicians dedicated to establish the Jackson and Bernstein-type theorems on the sphere see 1-9 . Early works such as Butzer and Johnen 3 Nikol skii and Lizorkin 8 9 and Lizorkin and Nikol skii 5 had successfully established the direct and inverse theorems on the sphere. In 1991 Li and Yang 4 constructed Jackson operators on the sphere and obtained the Jackson and Bernstein-type theorems for the Jackson operators. Jackson operator on the sphere is defined by see 4 1 Í
