Institute of Metrology and Computational Sciences, China Jiliang University, Hangzhou 310018, China
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 is constructed to approximate the function defined on the spherical cap . We thus establish the direct and inverse inequalities and obtain saturation theorems
for on the cap . Using methods of -functional and multiplier, we obtain the inequality
and that the saturation order of these operators is , where is the modulus of smoothness of degree 2, the constants and are independent of and .