The main purpose of this note is to observe that the celebrated theorem of Malliavin on the failure of spectral synthesis for the group algebra L{G) of a non compact locally compact abelian group G can be strengthened to get the failure of weak spectral synthesis (in the sense of Warner [7]) for £(G) and also for the "generalised group algebra" L{G, A). Let A be a commutative, semisimple, regul...

The metaplectic representation describes a class of automorphisms of the Heisenberg group H = H(G), defined for a locally compact abelian groupG. ForG = R ,H is the usual Heisenberg group. For the case when G is the finite cyclic group Zn, only partial constructions are known. Here we present new results for this case and we obtain an explicit construction of the metaplectic operators on C. We ...

An introduction to locally compact abelian algebraic groups and the usage of dual groups and Fourier transforms to study them.

We establish a regular sampling theory in the range of analysis operator continuous frame having unitary structure. The structure is related with representation locally compact abelian group on separable Hilbert space. samples are defined by means suitable discrete convolution systems which generalize some usual settings; here that taken at countable subgroup.

We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce a version of Sjöstrand’s class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since “hard analysis” techniques are not available on locally compact abelian groups, a new time-frequency appro...