Weyl quantization and a symbol calculus for abelian groups

Abstract

We develop a notion of a *-product on a general abelian group, establish a Weyl calculus for operators on the group and connect these with the representation theory of an associated Heisenberg group. This can all be viewed as a generalization of the familiar theory for R. A symplectic group is introduced and a connection with the classical Cayley transform is established. Our main application is to finite groups, where consideration of the symbol calculus for the cyclic groups provides an interesting alternative to the usual matrix form for linear transformations. This leads to a new basis for s\(n) and a decomposition of this Lie algebra into a sum of Cktan subalgebras.2000 Mathematics subject classification: primary 44A45 17B05 11F27.

Calculus of Operators: Covariant Transform and Relative Convolutions

“…Of course, by the Pontryagin duality Ĝ is isomorphic to G, in particular, T is isomorphic to Z. Our consideration of G shall be compared with [66].…”

“…Calculus of operators on groups and homogeneous spaces has a long history [13, 14, 17, 19, 22-24, 30-33, 36, 37, 54, 61] and is enjoying a recently revived interest [3,16,44,55,59,66]. There are some missing connections between two periods and the purpose of this presentation is to bridge the gap.…”

Gabor analysis over finite Abelian groups

“…These observations allow us to complement recent work of Wildberger on a symbolic calculus for finite abelian groups. In [49] the Weyl quantization was considered on finite abelian groups and one of the main results states that there is no good symbolic calculus for groups of even order. Our results do not rely on the order of Z N which is possible because we use a different kind of quantization, namely the Kohn-Nirenberg quantization.…”