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*

(1.51)

*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 ...

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