the existence of a weighted mean for almost periodic functions

in a recent paper by liang et al. (2010) , the original question which is that of the existence of a weighted mean for almost periodic functions was raised. in particular, they showed through an example that there exist weights for which a weighted mean for almost periodic functions may or may not exist. in this note we give some sufficient conditions which do guarantee the existence of a weighted mean for almost periodic functions, which will then coincide with the classical bohr mean. moreover, we will show that under those conditions, the corresponding weighted bohr transform exists.