# Difference between revisions of "Almost-period"

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− | A concept from the theory of almost-periodic functions (cf. [[Almost-periodic function|Almost-periodic function]]); a generalization of the notion of a period. For a uniformly almost-periodic function < | + | {{TEX|done}} |

+ | A concept from the theory of almost-periodic functions (cf. [[Almost-periodic function|Almost-periodic function]]); a generalization of the notion of a period. For a uniformly almost-periodic function $f(x)$, $-\infty<x<\infty$, a number $\tau=\tau_f(\epsilon)$ is called an $\epsilon$-almost-period of $f(x)$ if for all $x$, | ||

− | + | $$|f(x+\tau)-f(x)|<\epsilon.$$ | |

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

− | < | + | $$D_{S_l^p}[f(x+\tau),f(x)]<\epsilon,$$ |

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

− | A set of almost-periods of a function | + | A set of almost-periods of a function $f(x)$ is said to be relatively dense if there is a number $L=L(\epsilon,f)>0$ 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 $\epsilon$-almost-periods for these functions. |

====References==== | ====References==== | ||

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====Comments==== | ====Comments==== | ||

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

====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.S. Besicovitch, "Almost periodic functions" , Cambridge Univ. Press (1932)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.S. Besicovitch, "Almost periodic functions" , Cambridge Univ. Press (1932)</TD></TR></table> |

## Latest revision as of 13:17, 18 July 2014

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 $f(x)$, $-\infty<x<\infty$, a number $\tau=\tau_f(\epsilon)$ is called an $\epsilon$-almost-period of $f(x)$ if for all $x$,

$$|f(x+\tau)-f(x)|<\epsilon.$$

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

$$D_{S_l^p}[f(x+\tau),f(x)]<\epsilon,$$

where $D_{S_l^p}[f,\phi]$ is the distance between $f(x)$ and $\phi(x)$ 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 $L=L(\epsilon,f)>0$ 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 $\epsilon$-almost-periods for these functions.

#### References

[1] | B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian) |

#### Comments

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 $S_l^p$ $\epsilon$-periods. These characterizations are more complicated. A good additional reference is [a1], especially Chapt. II.

#### References

[a1] | A.S. Besicovitch, "Almost periodic functions" , Cambridge Univ. Press (1932) |

**How to Cite This Entry:**

Almost-period.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Almost-period&oldid=32494