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The $abc$-problem for Gabor systems


About this Title

Xin-Rong Dai and Qiyu Sun

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 244, Number 1152
ISBNs: 978-1-4704-2015-4 (print); 978-1-4704-3504-2 (online)
DOI: http://dx.doi.org/10.1090/memo/1152
Published electronically: June 17, 2016
Keywords:$abc$-problem for Gabor systems, Gabor frames, infinite matrices, piecewise linear transformation, ergodic theorem, sampling, shift-invariant spaces. \indent Xin-Rong Dai’s affiliation: School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China; email: daixr@mail.sysu.edu.cn. \indent Qiyu Sun’s affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816; email: qiyu.sun@ucf.edu.

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Table of Contents


Chapters

  • Preface
  • Chapter 1. Introduction
  • Chapter 2. Gabor Frames and Infinite Matrices
  • Chapter 3. Maximal Invariant Sets
  • Chapter 4. Piecewise Linear Transformations
  • Chapter 5. Maximal Invariant Sets with Irrational Time Shifts
  • Chapter 6. Maximal Invariant Sets with Rational Time Shifts
  • Chapter 7. The $abc$-problem for Gabor Systems
  • Appendix A. Algorithm
  • Appendix B. Uniform sampling of signals in a shift-invariant space

Abstract


A longstanding problem in Gabor theory is to identify time-frequency shifting lattices $a\mathbb Z×b\mathbb Z$ and ideal window functions $𝜒_{I}$ on intervals $I$ of length $c$ such that ${e^{-}2𝜋i n bt 𝜒_{I}(t- m a): (m, n)∈\mathbb Z×\mathbb Z}$ are Gabor frames for the space of all square-integrable functions on the real line. In this paper, we create a time-domain approach for Gabor frames, introduce novel techniques involving invariant sets of non-contractive and non-measure-preserving transformations on the line, and provide a complete answer to the above $abc$-problem for Gabor systems.

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