Editing Almost periodic function (section)

===Besicovitch almost periodic functions===
The space ''B''<sup>''p''</sup> of Besicovitch almost periodic functions was introduced by Besicovitch (1926).<ref>A.S. Besicovitch,   "On generalized almost periodic functions"  Proc. London Math. Soc. (2), 25  (1926)  pp.&nbsp;495–512</ref>
It is the closure of the trigonometric polynomials under the seminorm

:<math>\|f\|_{B,p}=\limsup_{x \to\infty}\left({1\over 2x} \int_{-x}^x |f(s)|^p \, ds \right)^{1/p}</math>

Warning: there are nonzero functions ''&fnof;'' with ||''&fnof;''||<sub>B,''p''</sub>&nbsp;=&nbsp;0, such as any bounded function of compact support, so to get a Banach space one has to quotient out by these functions.

The Besicovitch almost periodic functions in ''B''<sup>2</sup> have an expansion (not necessarily convergent) as

:<math>\sum a_ne^{i\lambda_n t}</math>

with Σ''a''{{supsub|2|''n''}} finite and ''λ''<sub>''n''</sub> real. Conversely every such series is the expansion of some Besicovitch periodic function (which is not unique).

The space ''B''<sup>''p''</sup> of Besicovitch almost periodic functions (for ''p''&nbsp;≥&nbsp;1) contains the space ''W''<sup>''p''</sup> of Weyl almost periodic functions. If one quotients out a subspace of "null" functions, it can be identified with the space of ''L''<sup>''p''</sup> functions on the Bohr compactification of the reals.