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, rho) is given and some applications of translation invariance of PAP(0)(X, rho) in PAP(X, rho) 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 rho (i.e. M(f, rho)=M(f(a), rho(a)) uniformly with respect to a is an element of R). Finally, the notion of ergodic zero set under a weight is introduced and its applications are exploited. (C) 2012 Elsevier Inc. All rights reserved.