Journal of Inequalities and Applications
Volume 2011 (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

Received 2 July 2010; Accepted 8 February 2011

Academic Editor: S. S. Dragomir

Copyright © 2011 Shaobo Lin 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.

Abstract

In the paper titled as “Jackson-type inequality on the sphere” (2004), Ditzian introduced a spherical nonconvolution operator 𝑂 𝑡 , 𝑟 , 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 𝐶 1 and 𝐶 2 such that 𝐶 1 𝜔 2 𝑟 ( 𝑓 , 𝑡 ) 𝑝 ≤ ‖ 𝑂 𝑡 , 𝑟 𝑓 − 𝑓 ‖ 𝑝 ≤ 𝐶 2 𝜔 2 𝑟 ( 𝑓 , 𝑡 ) 𝑝 for any 𝑝 th Lebesgue integrable or continuous function 𝑓 defined on the sphere, where 𝜔 2 𝑟 ( 𝑓 , 𝑡 ) 𝑝 is the 2 𝑟 th modulus of smoothness of 𝑓 .