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 Mat. Sb., 1994, Volume 185, Number 3, Pages 69–92 (Mi msb886)

The space of almost periodic functions with the Hausdorff metric

A. P. Petukhov

Abstract: The function space $\mathbf{H}$ obtained as the completion of the space $\mathbf{B}$ of real-valued uniformly almost periodic functions (a.p.) (Bohr a.p. functions) with respect to the Hausdorff metric is considered. Elements of the space $\mathbf{H}$ are called $H$-a.p. functions. Analogs of the theorems of Lyusternik (a criterion for compactness of a function family), Bochner (a criterion for almost periodicity), and Bohr (on representation of a.p. functions as diagonals of limit periodic functions) are obtained. The relationship between the space $\mathbf{H}$ and the space of $N$-a.p. functions is studied. In particular, it is shown that a continuous function in $\mathbf{H}$ may not belong to $\mathbf{B}$, but it is always an $N$-a.p. function. At the same time, the sum and the product of two continuous $H$-a.p. functions are not, in general, in $\mathbf{H}$ (but they are $N$-a.p. functions). Due to the coincidence of the topologies on $\mathbf{B}$ generated by the uniform and the Hausdorff metrics, the indicated space, in spite of its nonlinearity, is closer to the space $\mathbf{B}$ than the corresponding completions of $\mathbf{B}$ with respect to integral metrics.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 81:2, 321–341

Bibliographic databases:

UDC: 517.5
MSC: 42A75

Citation: A. P. Petukhov, “The space of almost periodic functions with the Hausdorff metric”, Mat. Sb., 185:3 (1994), 69–92; Russian Acad. Sci. Sb. Math., 81:2 (1995), 321–341

Citation in format AMSBIB
\Bibitem{Pet94} \by A.~P.~Petukhov \paper The space of almost periodic functions with the~Hausdorff metric \jour Mat. Sb. \yr 1994 \vol 185 \issue 3 \pages 69--92 \mathnet{http://mi.mathnet.ru/msb886} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1268798} \zmath{https://zbmath.org/?q=an:0834.42004} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1995 \vol 81 \issue 2 \pages 321--341 \crossref{https://doi.org/10.1070/SM1995v081n02ABEH003541} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RB51300004}