The abc-Problem for Gabor Systems (Memoirs of the American Mathematical Society)

The abc-Problem for Gabor Systems (Memoirs of the American Mathematical Society)

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  • 製本 Paperback:紙装版/ペーパーバック版／ページ数 99 p.
  • 言語 ENG
  • 商品コード 9781470420154
  • DDC分類 515.723

Full Description

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.

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