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 (It)t>0 acting on the submodule CP(G), p∈]1,+∞[, of the convolutionM1(G)-module LP(G) which consists of all functions f∈LP(G) admitting as their Fourier transformsFGf (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 Ĝ. Moreover, the relationship of the complex vector spaces CP(G) to some other function spaces, in particular to the vector spaces BP(G) introduced in [5], will be investigated.

Saturation on locally compact abelian groups: An extended theorem

/pdf/central-approximation-processes-26358ac1fh.pdf

Central approximation processes

http://link.springer.com/content/pdf/10.1007/978-94-010-1739-8_6.pdf

Saturationstheorie auf Kompakten Topologischen Gruppen

http://link.springer.com/content/pdf/10.1007/978-94-010-1739-8_18.pdf

Zur Theorie der Saturierten Approximationsverfahren auf Lokalkompakten Abelschen Gruppen

Spaces of test functions and theorems of the Bochner-Schoenberg-Eberlein type

Let G be an arbitrary locally compact abelian group. It is the purpose of the present paper to establish saturation theorems for approximation processes generated by families (μt)t > 0 of complex bounded Radon measures on G and operating on a submodule of the Banach module Lp(G), Lp(G), over the convolution algebra

. A basic tool is the Fourier transform method and, in the case p>1 for noncompact G, its interpretation in the context of the theory of quasimeasures on G.

Bernd Dreseler

Papers

Saturation on locally compact abelian groups

Saturation on locally compact abelian groups: An extended theorem