Saturation on locally compact abelian groups: An extended theorem

Let G be a locally compact abelian group. The concern of the present note is to extend (for exponents p>2) the saturation theorem on G stated as Theorem 4 in 5]. The extension will be established for approximation processes (I_t)_t>0 acting on the submodule C^P(G), p]1,+, of the convolutionM ¹(G)-module L^P(G) which consists of all functions fL^P(G) admitting as their Fourier transformsF _Gf (in the sense of the theory of quasimeasures) complex Radon measures not necessarily absolutely continuous with respect to any Haar measure on the dual group . Moreover, the relationship of the complex vector spaces C^P(G) to some other function spaces, in particular to the vector spaces B^P(G) introduced in 5], will be investigated.