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Vector-valued pseudo almost periodic functions. (English) Zbl 0901.42005

The author deals with a new generalization of (vector-valued) almost periodic functions to (vector-valued) pseudo almost periodic functions. Many well-known functions connected with the notion of almost periodic functions are contained as special cases in the class of these pseudo almost periodic functions.

Definition 5 defines a pseudo periodic function using a generalized almost periodicity and a generalized uniform continuity.

The pseudo almost periodic functions remind the former generalization of almost periodic functions to asymptotically almost periodic functions with their unique decomposition as a sum of an almost periodic function and a limit zero perturbation. The main result of this paper consists in Theorem 11 which shows that any pseudo almost periodic function has a unique decomposition as a sum of an almost periodic function and an ergodic perturbation.

The interesting Example 14 illustrates the fact that the generalized almost periodicity of a pseudo almost periodic function does not guarantee its generalized uniform continuity.

The space of pseudo almost periodic functions is a Banach space.

MSC:
42A75Classical almost periodic functions, mean periodic functions
43A60Almost periodic functions on groups, etc.; almost automorphic functions