Jackson-type inequalities on high-dimensional spheres

We study Jackson’s inequality on high-dimensional spheres with respect to the modulus of smoothness defined via the rotation group. We obtain a version of Jackson’s inequality with a dimension-free constant, extending Newman and Shapiro’s well-known results in 1964 from the case of r = 1 and p = ∞ to more general cases. Our results partially overcome the curse of dimensionality. We also establish similar results on the equivalence of the K-functional and modulus of smoothness.