A multiparameter semigroup of operators, mixed moduli and aproximation

This paper contains generalizations of results of N. P. Kuptsov and the author on approximation in a Banach space and of M. K. Potapov and the author on approximation of periodic functions of several variables.
Let multiparameter groups and a semigroup of commuting operators act in a Banach space. With respect to these there are defined mixed moduli for the elements of the space, analogues of the mixed moduli of smoothness for functions of several variables. With respect to the groups an approximation apparatus is constructed, generalizing the method which Potapov calls approximation by a “corner” and the author calls approximation by a mixed trigonometric polynomial. For approximations and mixed moduli, direct and inverse theorems are proved, and also propositions asserting that operators generating semigroups are integral powers of operators generating groups.
Bibliography: 15 items.