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 , 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 and such that for any th Lebesgue integrable or continuous function defined on the sphere, where is the th modulus of smoothness of .