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 $AP_\vf\rc$.