Almost-period

A concept from the theory of almost-periodic functions (cf. Almost-periodic function); a generalization of the notion of a period. For a uniformly almost-periodic function $f(x)$, $-\mathrm{\infty }<x<\mathrm{\infty }$, a number $\tau ={\tau }_f(ϵ)$ is called an $ϵ$-almost-period of $f(x)$ if for all $x$,

$$|f(x+\tau )-f(x)|<ϵ.$$

For generalized almost-periodic functions the concept of an almost-period is more complicated. For example, in the space $S_l^p$ an $ϵ$-almost-period $\tau$ is defined by the inequality

$$D_{S_l^p}[f(x+\tau ),f(x)]<ϵ,$$

where $D_{S_l^p}[f,\varphi ]$ is the distance between $f(x)$ and $\varphi (x)$ in the metric of $S_l^p$.

A set of almost-periods of a function $f(x)$ is said to be relatively dense if there is a number $L=L(ϵ,f)>0$ such that every interval $(\alpha ,\alpha +L)$ of the real line contains at least one number from this set. The concepts of uniformly almost-periodic functions and that of Stepanov almost-periodic functions may be defined by requiring the existence of relatively-dense sets of $ϵ$-almost-periods for these functions.

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For the definition of $S_l^p$ and its metric $D_{S_l^p}$ see Almost-periodic function. The Weyl, Besicovitch and Levitan almost-periodic functions can also be characterized in terms of $S_l^p$ $ϵ$-periods. These characterizations are more complicated. A good additional reference is [a1], especially Chapt. II.

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