In this section, definitions and main results on almost-periodic (AP) functions and their generalizations are presented for both continuous-and discrete-time cases. For extensive treatments on almost-periodic functions, see (Besicovitch 1932), (Bohr 1933), and (Corduneanu 1989) for continuous-time, and (Corduneanu 1989, Chapter VII), (Jessen and Tornehave 1945), and (von Neumann 1934) for discrete-time.
Definition 1.2.1 (Besicovitch 1932, Chapter 1). A function z(t), , is said to be uniformly almost-periodic if such that for any interval such that
The quantity is said translation number of z(t) corresponding to .
A set is said to be relatively dense in if such that the result is that D∩ I ≠ .
Thus, defined ...