Editing Almost periodic function (section)

===Uniform or Bohr or Bochner  almost periodic functions===

Bohr (1925)<ref>H. Bohr,   "Zur Theorie der fastperiodischen Funktionen I"  Acta Math., 45  (1925)  pp.&nbsp;29–127</ref> defined the '''uniformly almost-periodic functions''' as the closure of the trigonometric polynomials with respect to the [[uniform norm]] 
:<math>\|f\|_\infty = \sup_x|f(x)|</math>
(on [[bounded function]]s ''f'' on '''R''').  In other words, a function ''f'' is uniformly almost periodic if for every ''ε'' > 0 there is a finite linear combination of sine and cosine waves that is of distance less than ''ε'' from ''f'' with respect to the uniform norm.  The sine and cosine frequencies can be arbitrary real numbers.  Bohr [[mathematical proof|proved]] that this definition was equivalent to the existence of a [[relatively dense set]] of '''''ε''&nbsp;almost-periods''', for all ''ε''&nbsp;>&nbsp;0: that is, [[Translation (geometry)|translations]] ''T''(''ε'')&nbsp;=&nbsp;''T'' of the variable ''t''  making

:<math>\left|f(t+T)-f(t)\right|<\varepsilon.</math>

An alternative definition due to Bochner (1926) is equivalent to that of Bohr and is relatively simple to state:
<blockquote>A function ''f'' is almost periodic if every [[sequence]] {''&fnof;''(''t''&nbsp;+&nbsp;''T''<sub>''n''</sub>)} of translations of ''f'' has a [[subsequence]] that [[uniform convergence|converges uniformly]] for ''t'' in (&minus;&infin;,&nbsp;+&infin;).</blockquote>

The Bohr almost periodic functions are essentially the same as continuous functions on the [[Bohr compactification]] of the reals.