The Zak transform and representations induced from characters of an abelian subgroup

Abstract

We consider a variant of the Zak transform for a finite group G with respect to a fixed abelian subgroup H, and demonstrate a relationship with representations of G induced from characters of H. We also show how the Zak transform can be used to study right translations by H in L 2 (G), and give some examples of applications for equiangular tight frames.

Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration

“…Weil [43] defined the abstract Zak transform on arbitrary locally compact abelian (LCA) groups with respect to arbitrary closed subgroups. Subsequently, the Zak transform was reviewed and generalised by many authors, see for example [6,8,20,22,26]. Various connections between the Zak transform and (left) shift-invariant spaces were studied by several authors, cf.…”