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# 1.2 Almost-Periodic Functions

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.

## 1.2.1 Uniformly Almost-Periodic Functions

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 DI ≠ .

Thus, defined ...

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