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

Mahya Ghandehari, Kris Hollingsworth

Research output: Contribution to journalArticlepeer-review

Abstract

A discrete frame for L2(Rd) is a countable sequence {ej}j∈J in L2(Rd) together with real constants 0<A≤B<∞ such that A‖f‖22≤∑j∈J|〈f,ej〉|2≤B‖f‖22, for all f∈L2(Rd). 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 Mn(R)⋊GLn(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 languageEnglish (US)
Article number125328
JournalJournal of Mathematical Analysis and Applications
Volume503
Issue number2
DOIs
StatePublished - 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

Fingerprint

Dive into the research topics of 'Discrete frames for L<sup>2</sup>(R<sup>n<sup>2</sup></sup>) arising from tiling systems on GL<sub>n</sub>(R)'. Together they form a unique fingerprint.

Cite this