eBook ISBN:  9781470435042 
Product Code:  MEMO/244/1152.E 
List Price:  $79.00 
MAA Member Price:  $71.10 
AMS Member Price:  $47.40 
eBook ISBN:  9781470435042 
Product Code:  MEMO/244/1152.E 
List Price:  $79.00 
MAA Member Price:  $71.10 
AMS Member Price:  $47.40 

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 timefrequency 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 squareintegrable functions on the real line.
In this paper, the authors create a timedomain approach for Gabor frames, introduce novel techniques involving invariant sets of noncontractive and nonmeasurepreserving transformations on the line, and provide a complete answer to the above \(abc\)problem for Gabor systems.

Table of Contents

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 shiftinvariant space


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A longstanding problem in Gabor theory is to identify timefrequency 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 squareintegrable functions on the real line.
In this paper, the authors create a timedomain approach for Gabor frames, introduce novel techniques involving invariant sets of noncontractive and nonmeasurepreserving 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 shiftinvariant space