Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 205298, 16 pages
Research Article

The Direct and Converse Inequalities for Jackson-Type Operators on Spherical Cap

Institute of Metrology and Computational Sciences, China Jiliang University, Hangzhou 310018, China

Received 10 July 2009; Accepted 23 October 2009

Academic Editor: Marta A D García-Huidobro

Copyright © 2009 Yuguang Wang and Feilong 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.


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 {Jk,sm}k=1 is constructed to approximate the function defined on the spherical cap D(x0,γ). We thus establish the direct and inverse inequalities and obtain saturation theorems for {Jk,sm}k=1 on the cap D(x0,γ). Using methods of K-functional and multiplier, we obtain the inequality C1Jk,sm(f)fD,pω2(f,k1)D,pC2maxvkJv,sm(f)fD,p and that the saturation order of these operators is O(k2), where ω2(f,t)D,p is the modulus of smoothness of degree 2, the constants C1 and C2 are independent of k and f.