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.
UDC:
517.5
MSC: 42A75 Received: 17.10.1991 and 08.09.1992
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