We introduce the new class of Besicovitch-Musielak-Orlicz almost periodic functions and consider its strict convexity with respect to the Luxemburg norm.
2000 Mathematics Subject Classification: 44A35; 42A75; 47A16, 47L10, 47L80The Dunkl operators.* Supported by the Tunisian Research Foundation under 04/UR/15-02.
This paper is an extension of the work done in [Morsli M., Bedouhene F., Boulahia F., Duality properties and Riesz representation theorem in the Besicovitch-Orlicz space of almost periodic functions, Comment. Math. Univ. Carolin. 43 (2002), no. 1, 103--117] to the Besicovitch-Musielak-Orlicz space of almost periodic functions. Necessary and sufficient conditions for the reflexivity of this space are given. A Riesz type ``duality representation theorem'' is also stated.
In [6], the classical Riesz representation theorem is extended to the class of Besicovitch space of almost periodic functions a.p., . It is also shown that this space is reflexive. We shall consider here such results in the context of Orlicz spaces, namely in the class of Besicovitch-Orlicz space of almost periodic functions a.p., where is an Orlicz function.
The existence of mean periodic functions in the sense of L. Schwartz, generated, in various ways, by an equicontinuous group or an equicontinuous cosine function forces the spectral structure of the infinitesimal generator of or . In particular, it is proved under fairly general hypotheses that the spectrum has no accumulation point and that the continuous spectrum is empty.