This revised and expanded monograph presents the general theory for
frames and Riesz bases in Hilbert spaces as well as its concrete
realizations within Gabor analysis, wavelet analysis, and generalized
shift-invariant systems. Compared with the first edition, more emphasis
is put on explicit constructions with attractive properties. Based on
the exiting development of frame theory over the last decade, this
second edition now includes new sections on the rapidly growing fields
of LCA groups, generalized shift-invariant systems, duality theory for
as well Gabor frames as wavelet frames, and open problems in the field.
Key features include:
*Elementary introduction to frame theory in finite-dimensional spaces
* Basic results presented in an accessible way for both pure and
applied mathematicians
* Extensive exercises make the work suitable as a textbook for use in
graduate courses
* Full proofs includ
ed in introductory chapters; only basic knowledge of functional analysis
required
* Explicit constructions of frames and dual pairs of frames, with
applications and connections to time-frequency analysis, wavelets, and
generalized shift-invariant systems
* Discussion of frames on LCA groups and the concrete realizations in
terms of Gabor systems on the elementary groups; connections to sampling
theory
* Selected research topics presented with recommendations for more
advanced topics and further readin
g
* Open problems to stimulate further research
An Introduction to Frames and Riesz Bases will be of interest to
graduate students and researchers working in pure and applied
mathematics, mathematical physics, and engineering. Professionals
working in digital signal processing who wish to understand the theory
behind many modern signal processing tools may also find this book a
useful self-study reference.
Review of the first edition:
"Ole Christensen's An Introduction to Frames and Riesz Bases is a
first-rate introduction to the field ... . The book provides an
excellent exposition of these topics. The material is broad enough to
pique the interest of many readers, the included exercises supply some
interesting challenges, and the coverage provides enough background for
those new to the subject to begin conducting original research."
- Eric S. Weber, American Mathematical Monthly, Vol. 112, February,
2005
