Advances in Gabor Analysis

Rating:
Summary: A gem of a book
Review: This book contains a series of chapters which represent
a comprehensive analysis of some of the most important
topics in Gabor analysis as well as important new results
in these areas. The chapter on ``uncertainty principles''
finally brings the full scope of this important topic to
light - far beyond the ``classical uncertainty principle''.
One of the deepest questions in Gabor analysis is the
classification of the g,a,b so that (g,a,b) yields a Gabor
frame. Chapter 3 of this book gives a comprehensive
study of the case where g is the characteristic function
of an interval, showing that even this case is quite complicated
and not completely resolved yet. Operator Algebras are
playing an ever more important role in Gabor frame theory
and Chapter 6 gives one a glimpse into these powerful
techniques - especially in analyzing ``subspace Gabor frames''.
For applications, one needs to invert the frame operator.
These is a whole section in the book on the most up to
date results on inverting the Gabor frame operator.
One can also find other important gems in the book such as an
important introduction to pseudodifferential operators and
applications of Weyl-Heisenberg sets of stochastic encoding.
This is just a portion of the important topics in the book.
The chapters are very well written by leading experts in the
field, which allows one to ``get into'' a topic even with little
previous knowledge. Therefore, it is invaluable for people
who want to enter one of these areas of Gabor analysis for
the first time. Overall, this book is a significant contribution
to Gabor analysis and should sit on the desk of anyone working
in this area.

Rating:
Summary: A gem of a book
Review: This book contains a series of chapters which represent
a comprehensive analysis of some of the most important
topics in Gabor analysis as well as important new results
in these areas. The chapter on ``uncertainty principles''
finally brings the full scope of this important topic to
light - far beyond the ``classical uncertainty principle''.
One of the deepest questions in Gabor analysis is the
classification of the g,a,b so that (g,a,b) yields a Gabor
frame. Chapter 3 of this book gives a comprehensive
study of the case where g is the characteristic function
of an interval, showing that even this case is quite complicated
and not completely resolved yet. Operator Algebras are
playing an ever more important role in Gabor frame theory
and Chapter 6 gives one a glimpse into these powerful
techniques - especially in analyzing ``subspace Gabor frames''.
For applications, one needs to invert the frame operator.
These is a whole section in the book on the most up to
date results on inverting the Gabor frame operator.
One can also find other important gems in the book such as an
important introduction to pseudodifferential operators and
applications of Weyl-Heisenberg sets of stochastic encoding.
This is just a portion of the important topics in the book.
The chapters are very well written by leading experts in the
field, which allows one to ``get into'' a topic even with little
previous knowledge. Therefore, it is invaluable for people
who want to enter one of these areas of Gabor analysis for
the first time. Overall, this book is a significant contribution
to Gabor analysis and should sit on the desk of anyone working
in this area.