There are several different well studied moduli of smoothness on the unit sphere, including the classical one defined via the translation operators (i.e., averages over rims of spherical caps), the recent one introduced by Z. Ditzian via the group of rotations, and the most recent one introduced by Y. Xu and myself via finite order differences over Euler angles. In this talk, I will survey properties of these three different moduli of smoothness and some recent results related to them, such as the direct Jackson inequality and its Stechkin type inverse, the strong inverse inequality of type A and the equivalence with different K-functionals. I will also compare these moduli of smoothness and show that they are in fact equivalent in Lp spaces with 1 < p < ∞. Finally, I will discuss how these results on the sphere can be used to establish interesting results on algebraic polynomial approximation on the unit ball. In particular, I will discuss two new moduli of smoothness on the unit ball introduced by Y. Xu and myself, one of which is in analogy with the classical Ditzian-Totik moduli of smoothness on intervals.

Many of the results in this talk are from my joint works with Z. Ditzian and my joint works with Y. Xu.