| eBook ISBN: | 978-1-4704-3504-2 |
| Product Code: | MEMO/244/1152.E |
| List Price: | $79.00 |
| MAA Member Price: | $71.10 |
| AMS Member Price: | $47.40 |
| eBook ISBN: | 978-1-4704-3504-2 |
| Product Code: | MEMO/244/1152.E |
| List Price: | $79.00 |
| MAA Member Price: | $71.10 |
| AMS Member Price: | $47.40 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 244; 2016; 99 ppMSC: Primary 42; Secondary 37; 94
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 \(abc\)-problem for Gabor systems.
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Table of Contents
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Chapters
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Preface
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1. Introduction
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2. Gabor Frames and Infinite Matrices
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3. Maximal Invariant Sets
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4. Piecewise Linear Transformations
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5. Maximal Invariant Sets with Irrational Time Shifts
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6. Maximal Invariant Sets with Rational Time Shifts
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7. The $abc$-problem for Gabor Systems
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A. Algorithm
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B. Uniform sampling of signals in a shift-invariant space
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Additional Material
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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 \(abc\)-problem for Gabor systems.
-
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
