The abc-problem for Gabor systems
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 \{e^{-2\pi i n bt} \chi_I(t- m a):\ (m, n)\in \mathbb{Z}\times \mathbb{Z}\} are Gabor frames for the space of all square-integrable functions on the real line. In this paper, the authors create a time-domain approach for Gabor frames, introduce novel techniques involving invariant sets of non-contractive and n…
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- Full title :The abc-problem for Gabor systems / Xin-Rong Dai, Qiyu Sun.
Summary
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 \{e^{-2\pi i n bt} \chi_I(t- m a):\ (m, n)\in \mathbb{Z}\times \mathbb{Z}\} are Gabor frames for the space of all square-integrable functions on the real line. In this paper, the authors 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-pro.
Read More ↓Edition Statements
- Publication Date:©2016
Publication Statements
- Publisher(s):American mathematical Society
- Place of Publication:Providence, Rhode Island
- Language:eng
- Publication Date:©2016
Notes
"Volume 244, number 1152 (first of 4 numbers), November 2016."
Keywords: abc problem for Gabor systems, Gabor frames, infinite matrices, piecewise linear transformation, ergodic theorem, sampling, shift-invariant spaces.
The Physical Object
- Description:1 online resource (ix, 99 pages) : : illustrations
Edition Identifiers
- ISBN:9781470435042, 1470435047, 1470420155, 9781470420154
- oclc:ocn954720119
- LC Classification:QC20.7.W38
- Dewey Decimal:515/.2433
Table of Contents
Introduction -- Gabor frames and infinite matrices -- Maximal invariant sets -- Piecewise linear transformations -- Maximal invariant sets with irrational time shifts -- Maximal invariant sets with rational time shifts -- The abc-problem for Gabor systems -- Appendix A. Algorithm -- Appendix B. Uniform sampling of signals in a shift-invariant space -- Bibliography.
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