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 | ||
| − | + | $$\sum a_{n_1\dots n_k}e^{i(n_1\alpha_1+\ldots+n_k\alpha_k)x},$$ | |
| − | where | + | 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 |
| − | + | $$f(x)=f_1(x)+\ldots+f_k(x),$$ | |
| − | where each one of the functions | + | 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
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