Difference between revisions of "Bochner almost-periodic functions"
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| − | Functions equivalent to [[Bohr almost-periodic functions|Bohr almost-periodic functions]]; defined by S. Bochner [[#References|[1]]]. A function | + | {{TEX|done}} |
| + | Functions equivalent to [[Bohr almost-periodic functions|Bohr almost-periodic functions]]; defined by S. Bochner [[#References|[1]]]. A function $f(x)$ which is continuous in the interval $(-\infty,\infty)$ is said to be a Bochner almost-periodic function if the family of functions $\{f(x+h)\colon-\infty<h<\infty\}$ is compact in the sense of uniform convergence on $(-\infty,\infty)$, i.e. if it is possible to select from each infinite sequence $f(x+h_k)$, $k=1,2,\dots,$ a subsequence which converges uniformly to $f(x)$ on $(-\infty,\infty)$. Bochner's definition is extensively employed in the theory of almost-periodic functions; in particular, it serves as the starting point in abstract generalizations of the concept of almost-periodicity. | ||
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Revision as of 10:15, 24 August 2014
Functions equivalent to Bohr almost-periodic functions; defined by S. Bochner [1]. A function $f(x)$ which is continuous in the interval $(-\infty,\infty)$ is said to be a Bochner almost-periodic function if the family of functions $\{f(x+h)\colon-\infty<h<\infty\}$ is compact in the sense of uniform convergence on $(-\infty,\infty)$, i.e. if it is possible to select from each infinite sequence $f(x+h_k)$, $k=1,2,\dots,$ a subsequence which converges uniformly to $f(x)$ on $(-\infty,\infty)$. Bochner's definition is extensively employed in the theory of almost-periodic functions; in particular, it serves as the starting point in abstract generalizations of the concept of almost-periodicity.
References
| [1] | S. Bochner, "Beiträge zur Theorie der fastperiodischen Funktionen I, Funktionen einer Variablen" Math. Ann. , 96 (1927) pp. 119–147 |
| [2] | B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian) |
Comments
References
| [a1] | W. Maak, "Fastperiodische Funktionen" , Springer (1967) |
| [a2] | L. Amerio, G. Prouse, "Almost-periodic functions and functional equations" , v. Nostrand (1971) |
How to Cite This Entry:
Bochner almost-periodic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bochner_almost-periodic_functions&oldid=11522
Bochner almost-periodic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bochner_almost-periodic_functions&oldid=11522
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article