Weyl almost-periodic functions

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). The class was defined by H. Weyl [1]. The class $W^p$ of Weyl almost-periodic functions is an extension of the class of 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. [3].

References

[1]	H. Weyl, "Integralgleichungen und fastperiodische Funktionen" Math. Ann. , 97 (1927) pp. 338–356
[2]	B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian)
[3]	B.M. Levitan, V.V. Stepanov, "Sur les fonctions presque périodiques apportenant au sens strict à la classe " Dokl. Akad. Nauk SSSR , 22 : 5 (1939) pp. 220–223