View source for Weyl almost-periodic functions

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The class $W^p$ of complex-valued functions $f(x)$, $-\infty<x<\infty$, summable to degree $p$ on each bounded interval of the real axis and such that for every $\epsilon>0$ there is an $l=l(\epsilon,f)$ for which $f$ has a relatively-dense set $S_l^p$ of $\epsilon$-almost-periods (cf. [[Almost-period|Almost-period]]). The class was defined by H. Weyl [[#References|[1]]]. The class $W^p$ of Weyl almost-periodic functions is an extension of the class of [[Stepanov almost-periodic functions|Stepanov almost-periodic functions]].

Weyl almost-periodic functions are related to the metric

$$D_{W^p}(f,g)=\left\lbrace\lim_{l\to\infty}\sup_{-\infty<x<\infty}\frac{1}{2l}\int\limits_{x-l}^{x+l}|f(t)-g(t)|^pdt\right\rbrace^{1/p}.$$

If $\phi$ is a null function in the metric $D_{W^p}$, i.e.

$$\lim_{l\to\infty}\sup_x\frac{1}{2l}\int\limits_{x-l}^{x+l}|\phi(t)|^pdt=0,$$

and $f$ is a Stepanov almost-periodic function, then

$$f+\phi\tag{*}$$

is a Weyl almost-periodic function. There also exist Weyl almost-periodic functions which cannot be represented in the form \ref{*}; cf. [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Weyl,   "Integralgleichungen und fastperiodische Funktionen"  ''Math. Ann.'' , '''97'''  (1927)  pp. 338–356</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.M. Levitan,   "Almost-periodic functions" , Moscow  (1953)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.M. Levitan,   V.V. Stepanov,   "Sur les fonctions presque périodiques apportenant au sens strict à la classe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097680/w09768017.png" />"  ''Dokl. Akad. Nauk SSSR'' , '''22''' :  5  (1939)  pp. 220–223</TD></TR></table>