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| | ''uniform almost-periodic functions'' | | ''uniform almost-periodic functions'' |
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| − | 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 | + | 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 |
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| − | <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> | + | $$|f(x+\tau)-f(x)|<\epsilon,\quad-\infty<x<\infty.$$ |
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| − | 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 | + | 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 |
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| − | <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> | + | $$\inf_{-\infty<x<\infty}|g(x)|>\gamma>0.$$ |
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| − | 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. | + | 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. |
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| | ====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) |
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=15762
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article