AlmostPeriodicFunctionEquivalentDefinition 2005-07-10 01:50:45 2006-10-10 15:53:10 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here There is an equivalent definition of almost periodic function due to Bochner: A function $f \colon \mathbb{R} \to \mathbb{R}$ is \emph{almost periodic} if every sequence of translates of $f$ has a uniformly convergent subsequence. Not only is this definition simpler to state than that of Bohr, but it also generalizes to functions on groups.