The abc-Problem for Gabor Systems

ISBN-13: 9781470420154 / Angielski

Xin-Rong Dai;Qiyu Sun

Opis

A longstanding problem in Gabor theory is to identify time-frequency shifting lattices $amathbb imes bmathbb$ and ideal window functions $chi_I$ on intervals $I$ of length $c$ such that ${e^{-2pi i n bt} chi_I(t- m a): (m, n)in mathbb imes mathbb}$ 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.

Spis treści

• 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