Strong Converse Inequality for a Spherical Operator

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 wellknown 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 2r f, t p ≤ O t,r f − f p ≤ C 2 ω 2r f, t p for any pth Lebesgue integrable or continuous function f defined on the sphere, where ω 2r f, t p is the 2rth modulus of smoothness of f .