Translation invariance of weighted pseudo almost periodic functions and related problems

In this paper, some questions on weighted pseudo almost periodic functions are considered. Firstly, a sufficient condition for translation invariance of PAP 0 ( X , ρ ) is given and some applications of translation invariance of PAP 0 ( X , ρ ) in PAP ( X , ρ ) are presented. Then two mostly accounted types of weight are investigated and some results are given for ergodicity of an almost periodic function under a weight ρ (i.e. M ( f , ρ ) = M ( f a , ρ a ) uniformly with respect to a ∈ R ). Finally, the notion of ergodic zero set under a weight is introduced and its applications are exploited.