On Hartman almost periodic functions
We consider multi-dimensional Hartman almost periodic functions and sequences, defined with respect to different averaging sequences of subsets in $\mathbb R^d$ or $\mathbb Z^d$. We consider the behavior of their Fourier–Bohr coefficients and their spectrum, depending on the particular averaging sequence, and we demonstrate this dependence by several examples. Extensions to compactly generated, locally compact, abelian groups are considered. We define generalized Marcinkiewicz spaces based upon arbitrary measure spaces and general averaging sequences of subsets. We extend results of Urbanik to locally compact abelian groups.