---
tag: ED69H4C3PK
kind: concept
name: Bohr almost periodic function
msc: [42A75, 43A60]
specializes: [GZBSV8DYGS]
sources:
  - url: https://en.wikipedia.org/wiki/Almost_periodic_function
---

## Summary

A Bohr almost periodic function is a continuous function on the real line whose $\varepsilon$-almost periods are spread evenly over the line, for every $\varepsilon > 0$.
This was Harald Bohr's original definition of an [[almost periodic function|GZBSV8DYGS]].
These functions are exactly the uniform limits of trigonometric polynomials with arbitrary real frequencies, by [[Bohr's approximation theorem|GWCY1C1KQV]].
They have a mean value and a Fourier series, and they form a closed subalgebra of the bounded continuous functions.

## Explanations

### L1 human

A periodic function with period $p$ repeats exactly after every multiple of $p$.
A Bohr almost periodic function repeats only approximately, up to an error $\varepsilon$.
The shifts that achieve this are not equally spaced, but they never leave large gaps: every interval of some length $L$ contains one.
For a continuous periodic function one can take the multiples of the period, with $L$ equal to the period.