Strong Converse Inequality for a Spherical Operator

In the paper titled as “Jackson-type inequality on the sphere” 2004 , Ditzian introduced a spherical nonconvolution operator Ot,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 C1 and C2 such that C1ω2r f, t p ≤ ‖Ot,rf − f‖p ≤ C2ω2r f, t p for any pth Lebesgue integrable or continuous function f defined on the sphere, where ω2r f, t p is the 2rth modulus of smoothness of f .