---
tag: M702B26KQ8
kind: definition
name: Bohr almost periodic function
msc: [42A75, 43A60]
defines: ED69H4C3PK
uses: [0X3WGX95FE]
sources:
  - url: https://en.wikipedia.org/wiki/Almost_periodic_function
---

## Definition

Let $f \colon \R \to \C$ be a [[continuous function|0X3WGX95FE]].
For $\varepsilon > 0$, a number $\tau \in \R$ is an *$\varepsilon$-almost period* of $f$ if
$$\sup_{x \in \R} \abs{f(x + \tau) - f(x)} \le \varepsilon.$$
A set $T \subseteq \R$ is *relatively dense* if there is $L > 0$ such that every closed interval of length $L$ contains a point of $T$.
The function $f$ is *Bohr almost periodic* if for every $\varepsilon > 0$ the set of $\varepsilon$-almost periods of $f$ is relatively dense.

## Remarks

Every Bohr almost periodic function is bounded and uniformly continuous.
Bochner proved that $f$ is Bohr almost periodic if and only if every sequence of translates of $f$ has a uniformly convergent subsequence.
So the definition agrees with the general [[almost periodic function|GZBSV8DYGS]] on the group $\R$.

Example: let $f$ be continuous with period $p > 0$.
Every multiple $kp$, $k \in \Z$, satisfies $f(x + kp) = f(x)$, so it is an $\varepsilon$-almost period for every $\varepsilon > 0$.
Every closed interval of length $p$ contains a multiple of $p$, so $f$ is Bohr almost periodic.

Non-example: $f(x) = \arctan x$ is continuous, but not Bohr almost periodic.
For $\tau > 0$ the difference $\arctan(x + \tau) - \arctan x$ is largest at $x = -\tau/2$, where it equals $2 \arctan(\tau/2)$, and the case $\tau < 0$ is symmetric.
So for $0 < \varepsilon < \pi$ the $\varepsilon$-almost periods are the $\tau$ with $\abs{\tau} \le 2\tan(\varepsilon/2)$.
This set is bounded, hence not relatively dense.