Finite Dual g -Framelet Systems Associated with an Induced Group Action

Abstract

In this article, we first induce an action of a topological group G on ℓ2(ZNd) from a given action of G on the space C of complex numbers. Then, for each g ∈ G, we introduce a framelet system (g-framelet system or g-FS) associated with an induced action of G on ℓ2(ZNd), and a super g-FS for the super-space in the same set-up. By applying the group-theoretic approach based on the complete digit set, we characterize the generators of two g-framelet systems (super g-framelet systems) such that they form a g-dual pair (super g-dual pair). As a consequence, characterizations for the Parseval g-FS and the Parseval super g-FS are obtained. Further, some properties of the frame operator corresponding to the g-FS are observed, which results in concluding that its canonical dual preserves the same structure. © 2017, Springer International Publishing AG.