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Weyl almost-periodic functions

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The class 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
How to Cite This Entry:
Weyl almost-periodic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_almost-periodic_functions&oldid=33675
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article