#### How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

2. Complete and sign the license agreement.

3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2213  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

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

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.

View full volume PDF

View other years and numbers:

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.