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 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 2 r (f, t) p O t, r f - f p C 2 2 r (f, t) p for any p th Lebesgue integrable or continuous function f defined on the sphere, where 2 r (f, t) p is the 2 r th modulus of smoothness of f.