Translation generated oblique dual frames on locally compact groups

Abstract

Due to the redundancy property of frames, the stable decomposition of a vector in the separable Hilbert space (Formula presented.) allows the flexibility of choosing different types of duals for a frame. For a second countable locally compact group (Formula presented.) (not necessarily abelian) and a closed abelian subgroup Γ, we study the properties of oblique Γ-translation generated (Γ-TG) duals for a continuous frame in (Formula presented.). Two types of oblique Γ-TG duals viz., type-I and type-II are characterized in terms of the Zak transform for the pair (Formula presented.). Outside the group setup, first, we discuss such duals for the multiplication generated systems on the measure-theoretic abstraction in (Formula presented.) using the range function corresponding to the point-wise conditions in (Formula presented.), where X is a σ-finite measure space. Our results present a unified theory connecting the discrete problems with a continuous setup. Besides we characterize these duals' uniqueness using the Gramian/dual Gramian operators, which become a discrete frame/Riesz basis for the associated range space. As an application, we illustrate our results for (Formula presented.), p-adic numbers (Formula presented.) and locally compact abelian groups using fiberization map. © 2023 Informa UK Limited, trading as Taylor & Francis Group.