**Student Mathematical Library**

Volume: 40;
2007;
295 pp;
Softcover

MSC: Primary 47; 42; 15; 41;

Print ISBN: 978-0-8218-4212-6

Product Code: STML/40

List Price: $50.00

Individual Price: $40.00

**Electronic ISBN: 978-1-4704-2149-6
Product Code: STML/40.E**

List Price: $50.00

Individual Price: $40.00

#### Supplemental Materials

# Frames for Undergraduates

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*Deguang Han; Keri Kornelson; David Larson; Eric Weber*

A frame in a finite-dimensional inner-product space is a collection of
vectors spanning this space. In this sense, frames are generalizations of
orthonormal bases, which can be used in many cases where orthonormal bases
cannot be tailored to fit naturally arising applications. The wide array of
possibilities introduced by working with frames rather
than orthonormal bases has revolutionized mathematical areas such as wavelets
and harmonic analysis. The adaptability to existing conditions allows frames
to be used in applied settings including signal processing, imaging, sampling,
and cryptography.

The study of frames, particularly in finite dimensions, begins with exactly
the topics from an undergraduate linear algebra course. This makes the topics
particularly accessible to undergraduate students, yet the theory contains deep
unsolved problems. This book can be used as a resource for an REU or for a
topics course about frames. It is also a suitable textbook for a second linear
algebra course, using frames as a thematic example to demonstrate and explore
the new material.

The theory of frames is increasingly broad with widespread applications.
"Frames for Undergraduates" introduces students to this vibrant and important
area of mathematics.

#### Readership

Undergraduate and graduate students interested in linear algebra and applications, and the theory of frames.

#### Reviews & Endorsements

This incredibly readable book is an enticing introduction to frames. It contains the perfect combination of fundamental ideas from linear algebra and operator theory, and foundational examples and applications of finite frames. My students and I have found it to be an all-around pleasure to read.

-- Michael Orrison, Harvey Mudd College

I used this book in an undergraduate special topics course on frame theory. With the prerequisite being a standard linear algebra course, I appreciated the comprehensive yet economical background on the necessary ideas from linear algebra and finite dimensional operator theory. The chapters on frame theory cover a nice range of topics, from sampling theory and image reconstruction to frames arising from unitary representations of a group to more recent developments like frame potential, which has interesting connections to physics and platonic solids. My students were inspired not only by these topics but also by the final chapter on anecdotes; many of them had never entertained the possibility that they could do original mathematical research as undergraduates.

-- Fumiko Futamura, Ph.D., Southwestern University

The result (of this book) is a concise and accessible treatment that serves well as a semester's text.

-- Mathematical Horizon

This book would be a good candidate for a topics course or for a second course in linear algebra.

-- MAA Reviews

the clean conceptual layout of this little text makes it appropriate for a second semester special topics course in linear algebra for mathematics majors and certainly for directed reading for a junior level student who might be contemplating graduate study in mathematics. Exercises are included.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Frames for Undergraduates

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface for Instructors ix10 free
- Acknowlegements xiii14 free
- Introduction 116 free
- Chapter 1. Linear Algebra Review 520 free
- §1.1. Vector Spaces 520
- §1.2. Bases for Vector Spaces 823
- §1.3. Linear Operators and Matrices 1732
- §1.4. The Rank of a Linear Operator and a Matrix 2338
- §1.5. Determinant and Trace 2540
- §1.6. Inner Products and Orthonormal Bases 2742
- §1.7. Orthogonal Direct Sum 3449
- §1.8. Exercises from the Text 3651
- §1.9. Additional Exercises 3752

- Chapter 2. Finite-Dimensional Operator Theory 4156
- §2.1. Linear Functionals and the Dual Space 4156
- §2.2. Riesz Representation Theorem and Adjoint Operators 4358
- §2.3. Self-adjoint and Unitary Operators 4661
- §2.4. Orthogonal Complements and Projections 5267
- §2.5. The Moore-Penrose Inverse 5974
- §2.6. Eigenvalues for Operators 6075
- §2.7. Square Roots of Positive Operators 6782
- §2.8. The Polar Decomposition 6984
- §2.9. Traces of Operators 7287
- §2.10. The Operator Norm 7489
- §2.11. The Spectral Theorem 7792
- §2.12. Exercises from the Text 8196
- §2.13. Additional Exercises 8297

- Chapter 3. Introduction to Finite Frames 87102
- §3.1. R[sup(n)]-Frames 88103
- §3.2. Parseval Frames 92107
- §3.3. General Frames and the Canonical Reconstruction Formula 98113
- §3.4. Frames and Matrices 104119
- §3.5. Similarity and Unitary Equivalence of Frames 109124
- §3.6. Frame Potential 113128
- §3.7. Numerical Algorithms 118133
- §3.8. Exercises from the Text 121136
- §3.9. Additional Exercises 121136

- Chapter 4. Frames in R[sup(2)] 123138
- §4.1. Diagram Vectors 123138
- §4.2. Equivalence of Frames 125140
- §4.3. Unit Tight Frames with Four Vectors 129144
- §4.4. Unit Tight Frames with k Vectors 131146
- §4.5. Fundamental Inequality in R[sup(2)] 134149
- §4.6. Frame Surgery: Removals and Replacements 137152
- §4.7. Exercises from the Text 138153
- §4.8. Additional Exercises 139154

- Chapter 5. The Dilation Property of Frames 141156
- Chapter 6. Dual and Orthogonal Frames 155170
- Chapter 7. Frame Operator Decompositions 183198
- Chapter 8. Harmonic and Group Frames 205220
- Chapter 9. Sampling Theory 229244
- Chapter 10. Student Presentations 265280
- Chapter 11. Anecdotes: Frame Theory Projects by Undergraduates 281296
- Bibliography 287302
- Index of Symbols 291306 free
- Index 293308
- Back Cover Back Cover 1314