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''uniform almost-periodic functions''
 
''uniform almost-periodic functions''
  
The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b0167701.png" />-a.-p. of almost-periodic functions. The first definition, which was given by H. Bohr [[#References|[1]]], is based on a generalization of the concept of a period: A continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b0167702.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b0167703.png" /> is a Bohr almost-periodic function if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b0167704.png" /> there exists a relatively-dense set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b0167705.png" />-almost-periods of this function (cf. [[Almost-period|Almost-period]]). In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b0167706.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b0167707.png" />-almost-periodic (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b0167708.png" />-a.-p.) if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b0167709.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677010.png" /> such that in each interval of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677011.png" /> there exists at least one number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677012.png" /> such that
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The class $U$-a.-p. of almost-periodic functions. The first definition, which was given by H. Bohr [[#References|[1]]], is based on a generalization of the concept of a period: A continuous function $f(x)$ on the interval $(-\infty,\infty)$ is a Bohr almost-periodic function if for any $\epsilon>0$ there exists a relatively-dense set of $\epsilon$-almost-periods of this function (cf. [[Almost-period|Almost-period]]). In other words, $f(x)$ is $U$-almost-periodic (or $\in U$-a.-p.) if for any $\epsilon>0$ there exists an $L=L(\epsilon)$ such that in each interval of length $L$ there exists at least one number $\tau$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677013.png" /></td> </tr></table>
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$$|f(x+\tau)-f(x)|<\epsilon,\quad-\infty<x<\infty.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677014.png" />, is bounded, a Bohr almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677015.png" /> becomes a continuous periodic function. Bochner's definition (cf. [[Bochner almost-periodic functions|Bochner almost-periodic functions]]), which is equivalent to Bohr's definition, is also used in the theory of almost-periodic functions. Functions in the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677017.png" />-almost-periodic functions are bounded and uniformly-continuous on the entire real axis. The limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677018.png" /> of a uniformly-convergent sequence of Bohr almost-periodic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677019.png" /> belongs to the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677020.png" />-almost-periodic functions; this class is invariant with respect to arithmetical operations (in particular the Bohr almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677021.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677022.png" />-almost-periodic, under the condition
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If $L(\epsilon),\epsilon\to0$, is bounded, a Bohr almost-periodic function $f(x)$ becomes a continuous periodic function. Bochner's definition (cf. [[Bochner almost-periodic functions|Bochner almost-periodic functions]]), which is equivalent to Bohr's definition, is also used in the theory of almost-periodic functions. Functions in the class of $U$-almost-periodic functions are bounded and uniformly-continuous on the entire real axis. The limit $f(x)$ of a uniformly-convergent sequence of Bohr almost-periodic functions $\{f_n(x)\}$ belongs to the class of $U$-almost-periodic functions; this class is invariant with respect to arithmetical operations (in particular the Bohr almost-periodic function $f(x)/g(x)$ is $U$-almost-periodic, under the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677023.png" /></td> </tr></table>
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$$\inf_{-\infty<x<\infty}|g(x)|>\gamma>0.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677024.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677025.png" />-almost-periodic and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677026.png" /> is uniformly continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677028.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677029.png" />-almost-periodic; the indefinite integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677030.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677031.png" />-almost-periodic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016770/b01677032.png" /> is a bounded function.
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If $f(x)$ is $U$-almost-periodic and if $f'(x)$ is uniformly continuous on $(-\infty,\infty)$, then $f'(x)$ is $U$-almost-periodic; the indefinite integral $F(x)=\int_0^xf(t)dt$ is $U$-almost-periodic if $F(x)$ is a bounded function.
  
 
====References====
 
====References====

Latest revision as of 10:22, 24 August 2014

uniform almost-periodic functions

The class $U$-a.-p. of almost-periodic functions. The first definition, which was given by H. Bohr [1], is based on a generalization of the concept of a period: A continuous function $f(x)$ on the interval $(-\infty,\infty)$ is a Bohr almost-periodic function if for any $\epsilon>0$ there exists a relatively-dense set of $\epsilon$-almost-periods of this function (cf. Almost-period). In other words, $f(x)$ is $U$-almost-periodic (or $\in U$-a.-p.) if for any $\epsilon>0$ there exists an $L=L(\epsilon)$ such that in each interval of length $L$ there exists at least one number $\tau$ such that

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

If $L(\epsilon),\epsilon\to0$, is bounded, a Bohr almost-periodic function $f(x)$ becomes a continuous periodic function. Bochner's definition (cf. Bochner almost-periodic functions), which is equivalent to Bohr's definition, is also used in the theory of almost-periodic functions. Functions in the class of $U$-almost-periodic functions are bounded and uniformly-continuous on the entire real axis. The limit $f(x)$ of a uniformly-convergent sequence of Bohr almost-periodic functions $\{f_n(x)\}$ belongs to the class of $U$-almost-periodic functions; this class is invariant with respect to arithmetical operations (in particular the Bohr almost-periodic function $f(x)/g(x)$ is $U$-almost-periodic, under the condition

$$\inf_{-\infty<x<\infty}|g(x)|>\gamma>0.$$

If $f(x)$ is $U$-almost-periodic and if $f'(x)$ is uniformly continuous on $(-\infty,\infty)$, then $f'(x)$ is $U$-almost-periodic; the indefinite integral $F(x)=\int_0^xf(t)dt$ is $U$-almost-periodic if $F(x)$ is a bounded function.

References

[1] H. Bohr, "Zur Theorie der fastperiodischen Funktionen I" Acta Math. , 45 (1925) pp. 29–127
[2] B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian)


Comments

Bohr's treatise [a1] is a good reference. An up-to-date reference is [a2].

References

[a1] H. Bohr, "Almost periodic functions" , Chelsea, reprint (1947) (Translated from German)
[a2] C. Corduneanu, "Almost periodic functions" , Wiley (1968)
How to Cite This Entry:
Bohr almost-periodic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohr_almost-periodic_functions&oldid=33123
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article