**Proceedings of Symposia in Applied Mathematics**

Volume: 73;
2016;
264 pp;
Hardcover

MSC: Primary 15; 41; 42; 47; 52; 65; 90;

Print ISBN: 978-1-4704-2019-2

Product Code: PSAPM/73

List Price: $99.00

Individual Member Price: $79.20

**Electronic ISBN: 978-1-4704-3212-6
Product Code: PSAPM/73.E**

List Price: $99.00

Individual Member Price: $79.20

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# Finite Frame Theory: A Complete Introduction to Overcompleteness

Share this page *Edited by *
*Kasso A. Okoudjou*

Frames are overcomplete sets of vectors that
can be used to stably and faithfully decompose and reconstruct vectors
in the underlying vector space. Frame theory stands at the
intersection of many areas in mathematics such as functional and
harmonic analysis, numerical analysis, matrix theory, numerical linear
algebra, algebraic and differential geometry, probability, statistics,
and convex geometry. At the same time its applications in
engineering, medicine, computer science, and quantum computing are
motivating new research problems in applied and pure mathematics.

This volume is based on lectures delivered at the 2015 AMS Short
Course “Finite Frame Theory: A Complete Introduction to
Overcompleteness”, held January 8–9, 2015 in San Antonio,
TX.

Mostly written in a tutorial style, the seven chapters contained in
this volume survey recent advances in the theory and applications of
finite frames. In particular, it presents state-of-the-art results on
foundational frame problems, and on the analysis and design of various
frames, mostly motivated by specific applications. Carefully
assembled, the volume quickly introduces the non-expert to the basic
tools and techniques of frame theory. It then moves to develop many
recent results in the area and presents some important
applications. As such, the volume is designed for a diverse audience
including researchers in applied and computational harmonic analysis,
as well as engineers and graduate students.

#### Readership

Graduate students and research mathematicians interested in frame theory and its applications.

# Table of Contents

## Finite Frame Theory: A Complete Introduction to Overcompleteness

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Introduction ix10
- Bibliography xiii14
- A Brief Introduction to Hilbert Space Frame Theory and its Applications 116
- 1. Reading List 116
- 2. The Basics of Hilbert Space Theory 217
- 3. The Basics of Operator Theory 621
- 4. Hilbert Space Frames 1631
- 5. Constants Related to Frames 2843
- 6. Constructing Finite Frames 2944
- 7. Gramian Operators 3550
- 8. Fusion Frames 3651
- 9. Infinite Dimensional Hilbert Spaces 3853
- 10. Major Open Problems in Frame Theory 4055
- References 4863

- Unit norm tight frames in finite-dimensional spaces 5368
- Algebro-Geometric Techniques and Geometric Insights for Finite Frames 7994
- Preconditioning techniques in frame theory and probabilistic frames 105120
- Quantization, finite frames, and error diffusion 143158
- Frames and Phaseless Reconstruction 175190
- Compressed sensing and dictionary learning 201216
- Index 243258
- Back Cover Back Cover1266