# memo_has_moved_text();The $abc$-problem for Gabor systems

Xin-Rong Dai, School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China and Qiyu Sun, Department of Mathematics, University of Central Florida, Orlando, Florida 32816

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: https://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.
MSC: Primary 42C15, 42C40; Secondary 37A05, 94A20

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Chapters

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

### Abstract

A longstanding problem in Gabor theory is to identify time-frequency shifting lattices $a\mathbb {Z}\times b\mathbb {Z}$ and ideal window functions $\chi _I$ on intervals $I$ of length $c$ such that $\left \{\,e^{-2\pi i n bt} \chi _I(t- m a): (m, n)\in \mathbb {Z}\times \mathbb {Z}\,\right \}$ 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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