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Difference between revisions of "Bohl almost-periodic functions"

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A class of functions the typical property of which is that they can be uniformly approximated on the whole real axis by generalized trigonometric polynomials of the form
 
A class of functions the typical property of which is that they can be uniformly approximated on the whole real axis by generalized trigonometric polynomials of the form
  
<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/b016760/b0167601.png" /></td> </tr></table>
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$$\sum a_{n_1\dots n_k}e^{i(n_1\alpha_1+\ldots+n_k\alpha_k)x},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016760/b0167602.png" /> are arbitrary integers, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016760/b0167603.png" /> are given real numbers. This class of functions contains the class of continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016760/b0167604.png" />-periodic functions and is contained in the class of [[Bohr almost-periodic functions|Bohr almost-periodic functions]]. P. Bohl specified several necessary and sufficient conditions for a function to be almost-periodic. In particular, any function of the type
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where $n_1,\dots,n_k$ are arbitrary integers, while $\alpha_1,\dots,\alpha_k$ are given real numbers. This class of functions contains the class of continuous $2\pi$-periodic functions and is contained in the class of [[Bohr almost-periodic functions|Bohr almost-periodic functions]]. P. Bohl specified several necessary and sufficient conditions for a function to be almost-periodic. In particular, any function of the type
  
<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/b016760/b0167605.png" /></td> </tr></table>
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$$f(x)=f_1(x)+\ldots+f_k(x),$$
  
where each one of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016760/b0167606.png" /> is continuous and periodic (with possibly different periods), is a Bohl almost-periodic function.
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where each one of the functions $f_1(x),\dots,f_k(x)$ is continuous and periodic (with possibly different periods), is a Bohl almost-periodic function.
  
 
====References====
 
====References====

Revision as of 07:23, 28 September 2014

A class of functions the typical property of which is that they can be uniformly approximated on the whole real axis by generalized trigonometric polynomials of the form

$$\sum a_{n_1\dots n_k}e^{i(n_1\alpha_1+\ldots+n_k\alpha_k)x},$$

where $n_1,\dots,n_k$ are arbitrary integers, while $\alpha_1,\dots,\alpha_k$ are given real numbers. This class of functions contains the class of continuous $2\pi$-periodic functions and is contained in the class of Bohr almost-periodic functions. P. Bohl specified several necessary and sufficient conditions for a function to be almost-periodic. In particular, any function of the type

$$f(x)=f_1(x)+\ldots+f_k(x),$$

where each one of the functions $f_1(x),\dots,f_k(x)$ is continuous and periodic (with possibly different periods), is a Bohl almost-periodic function.

References

[1] P. Bohl, "Über die Darstellung von Funktionen einer Variabeln durch trigonometrische Reihen mit mehreren einer Variabeln proportionalen Argumenten" , Dorpat (1893) (Thesis)
[2] P. Bohl, "Ueber eine Differentialgleichung der Störungstheorie" J. Reine Angew. Math. , 131 (1906) pp. 268–321
[3] B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian)


Comments

A very well-known reference for this kind of topic is [a1].

References

[a1] H. Bohr, "Almost periodic functions" , Chelsea, reprint (1947) (Translated from German)
How to Cite This Entry:
Bohl almost-periodic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohl_almost-periodic_functions&oldid=16534
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article