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 $\mathcal{H}$ allows the flexibility of choosing different types of duals for a frame. For a second countable locally compact group $\mathcal{G}$ (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 $L^2(\mathcal{G})$. Two types of oblique Γ-TG duals viz., type-I and type-II are characterized in terms of the Zak transform for the pair $(\mathcal{G},\mathrm{\Gamma })$. Outside the group setup, first, we discuss such duals for the multiplication generated systems on the measure-theoretic abstraction in $L^2(X;\mathcal{H})$ using the range function corresponding to the point-wise conditions in $\mathcal{H}$, 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 ${\mathbb{R}}^n$, p-adic numbers ${\mathbb{Q}}_p$ and locally compact abelian groups using fiberization map.

Keywords:

• Locally compact group
• translation invariant space
• multiplication invariant space
• dual frame
• Zak transformation

2000 Mathematics Subject Classifications:

• 42C15
• 43A32
• 47A15
• 43A65
• 46C05

Acknowledgments

The authors are grateful to the referee for meticulously reading the manuscript and providing several valuable suggestions in revising the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

Research of S. Sarkar and N. K. Shukla was supported by research grant from CSIR, New Delhi [grant number 09/1022(0037)/2017-EMR-I] and NBHM-DAE [grant number 02011/19/2018-NBHM(R.P.)/R&D II/14723], respectively. The authors also acknowledge the facilities of the Bhaskaracharya Mathematics Laboratory, IIT Indore, supported by the DST-FIST Project [file number SR/FST/MS I/2018/26].