Translation invariance of weighted pseudo almost periodic functions and related problems (Q412435)

The authors study weighted pseudo almost periodic functions denoted by \(PAP (\mathbb {X},\rho)\) \(=PAP_0 (\mathbb {X},\rho)+ AP (\mathbb {R},\mathbb {X}) \), where \(PAP_0 (\mathbb {X},\rho)\) is the set of all bounded continuous functions with vanishing mean value under weight \(\rho\). A sufficient condition for translation invariance of \(PAP_0 (\mathbb {X},\rho)\) is given and some applications of translation invariance of \(PAP_0 (\mathbb {X},\rho)\) in \(PAP (\mathbb {X},\rho)\) are presented. Two examples of weights are investigated and some results are given for ergodicity of an almost periodic function under a weight \(\rho\). Finally, the notion of ergodic zero set under a weight is introduced and its applications are exploited.