An Introduction to Frames and Riesz Bases

  • Ole Christensen

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Table of contents

  1. Front Matter
    Pages i-xxv
  2. Ole Christensen
    Pages 67-108
  3. Ole Christensen
    Pages 109-118
  4. Ole Christensen
    Pages 119-151
  5. Ole Christensen
    Pages 153-164
  6. Ole Christensen
    Pages 165-181
  7. Ole Christensen
    Pages 183-198
  8. Ole Christensen
    Pages 199-240
  9. Ole Christensen
    Pages 241-256
  10. Ole Christensen
    Pages 257-286
  11. Ole Christensen
    Pages 287-325
  12. Ole Christensen
    Pages 327-357
  13. Ole Christensen
    Pages 407-415
  14. Ole Christensen
    Pages 417-444
  15. Ole Christensen
    Pages 445-478
  16. Ole Christensen
    Pages 479-492

About this book

Introduction

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 

Keywords

Frame Theory Riesz Bases Vector Spaces Hilbert Spaces Gabor Frames Wavelets LCA Groups Generalized Shift-invariant Systems

Authors and affiliations

  • Ole Christensen
    • 1
  1. 1.Dept Math and Comp Sci,Building 303Technical University of DenmarkLyngbyDenmark

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-25613-9
  • Copyright Information Springer International Publishing Switzerland 2016
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-25611-5
  • Online ISBN 978-3-319-25613-9
  • Series Print ISSN 2296-5009
  • Series Online ISSN 2296-5017
  • About this book
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