New moduli of smoothness on the unit ball, applications and computability

In a previous article I introduced new moduli of smoothness on the unit ball and derived their main properties. The usefulness of these moduli is demonstrated here by some applications. Ul’yanov-type inequalities are derived via new Nikolski-type inequalities. Sharp Jackson inequalities are given for some Banach spaces whose unit ball satisfies a certain convexity condition. Connections with appropriate Besov spaces are discussed. Computations of the moduli for some functions are exhibited. This will illustrate the situations and the optimality of the conditions in some theorems as well as demonstrate the computability of the moduli.