... p<∞.1.3In the following, Lpd−1 will always be one of the spaces Lpd−1 for 1 ≤ p<∞,orCd−1 for p ∞. For an arbitrary number θ,0<θ<π, we define the spherical translation operator ... well-knownJackson type inequality for spherical harmonics. Before giving the definition of Ot,r, we needto introduce some preliminaries. DenoteTρfx: fρx for ρ ∈ SOd,x∈d−1,1.9where ... inequality for Poisson sums,” Proceedings of the AmericanMathematical Society, vol. 133, no. 9, pp. 2609–2611, 2005.9 R. Y. Yang, F. L. Cao, and J. Y. Xiong, “The strong converse inequalities for...