Some spaces of almost periodic functions

Abstract

The oscillatory functions are sums of generalized
Fourier or tri\-go\-no\-me\-tric series of the form
$\sum\limits^\9_{k=1} a_k\exp[i\lbb_k(t)],$ with $a_k\in\calc$ and
\mbox{ $\lbb_k:R\to R$,} $k\ge1$, some generalized exponents. The
class of almost periodic functions, as those defined by H.~Bohr
(1923-25) and generalized by Stepanov, Besicovitch, Weyl a.o.,
corresponds to the choice of exponents given by $\lbk(t)=\lbk t,$
$\lbk\in R,$ $t\in R,$ $k\ge1.$ We shall consider, in this paper,
a generalization of almost periodicity which is described by the
spaces $AP_r\rc$, $1\le r\le2$, as shown in our papers \cite{c1},
\cite{c2}, \cite{c3}. See, also, Shubin \cite{s1}, \cite{s2},
Osipov \cite{o1} and Zhang \cite{z1}, \cite{z2}, \cite{z3}, as
well as the book Corduneanu \cite{c4}.

The new class of almost periodic functions, we want to construct
in this paper, is dependent of the choice of a function
$\vf:R_+\to R_+,$ with the properties: $\vf(0)=0$, $\vf(r)>0$ for
$r>0$ and increasing. This class of functions, very oftenly used
in the theory of stability, bound on Liapunov's functions or
functionals, is known as Kamke's type functions. We shall consider
only continuous functions in this class, denoted by $K$, which
assures the existence of the inverse $\vf\1\in K$, as well as the
continuity of $\vf\1$ when $\vf$ is continuous.

The spaces we will construct will be denoted by $AP_\vf\rc$,
consisting of maps from $R$ into $\calc$. Further conditions shall
be added on the function $\vf$, in order to assure a mixed
algebraic-topological structure for the space $AP_\vf\rc$.