Discrete frames for L2(Rn2) arising from tiling systems on GLn(R)

Mahya Ghandehari; Kris Hollingsworth

doi:10.1016/j.jmaa.2021.125328

Discrete frames for L²(R^n²) arising from tiling systems on GL_n(R)

Mahya Ghandehari, Kris Hollingsworth

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Abstract

A discrete frame for L²(R^d) is a countable sequence {e_j}_j∈J in L²(R^d) together with real constants 0<A≤B<∞ such that A‖f‖₂²≤∑j∈J|〈f,e_j〉|²≤B‖f‖₂², for all f∈L²(R^d). We present a method of sampling continuous frames, which arise from square-integrable representations of affine-type groups, to create discrete frames for high-dimensional signals. Our method relies on partitioning the ambient space by using a suitable “tiling system”. We provide all relevant details for constructions in the case of M_n(R)⋊GL_n(R), although the methods discussed here are general and could be adapted to some other settings. Finally, we prove significantly improved frame bounds over the previously known construction for the case of n=2.

Original language	English (US)
Article number	125328
Journal	Journal of Mathematical Analysis and Applications
Volume	503
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2021.125328
State	Published - Nov 15 2021

Bibliographical note

Funding Information:
This project was initiated during a summer research program funded by University of Delaware Graduate Program Improvement and Innovation Grants “GEMS”. The second author thanks University of Delaware for funding his GEMS project in summer 2016 and summer 2018. The first author was partially supported by University of Delaware Research Foundation , and partially by NSF grant DMS-1902301 , while this work was being completed. She also thanks the Department of Mathematical Sciences at Delaware for its support while important revisions were made to the paper. We sincerely thank the anonymous reviewer for critically reading the manuscript and suggesting substantial improvements. Additionally we would like to thank Karlheinz Groechenig for several helpful remarks on an early draft and providing information about historical context of our work. Finally, the authors would like to thank Nathaniel Kim and Paige Shumskas for proofreading early drafts of this work.

Publisher Copyright:
© 2021 Elsevier Inc.

Keywords

Continuous wavelet transform
Discrete frame
Quasiregular representation
Square-integrable representation
Tiling system

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