New Moduli of Smoothness on the Unit Ball and Other Domains, Introduction and Main Properties

A new set of moduli of smoothness on a large variety of Banach spaces of functions on the unit ball is introduced. These measures of smoothness utilize uniformly bounded holomorphic semigroups on the Banach space in question. The new moduli are "correct" in the sense that they satisfy direct (Jackson) and weak converse inequalities. The method used also applies to spaces of functions on the simplex and the unit sphere, and while the main goal is the investigation of properties and relations concerning the unit ball, many of the results will be given for other domains and situations. The classic properties, including equivalence with appropriate -functionals or realization functionals, will be established. Bernstein- and Kolmogorov-type inequalities are proved.