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

In this article, we first induce an action of a topological group G on l(2)(Z(N)(d)) from a given action of G on the space C of complex numbers. Then, for each g is an element of G, we introduce a framelet system (g-framelet system or g-FS) associated with an induced action of G on l(2)(Z(N)(d)), 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.