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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.

Deguang Han: University of Central Florida, Orlando, FL,

Keri Kornelson: Grinnell College, Grinnell, IA,

David Larson: Texas A&M University, College Station, TX,

Eric Weber: Iowa State University, Ames, IA

Cover 1
Title 4
Copyright 5
Contents 6
Preface for Instructors 10
Acknowlegements 14
Introduction 16
Chapter 1. Linear Algebra Review 20
§1.1. Vector Spaces 20
§1.2. Bases for Vector Spaces 23
§1.3. Linear Operators and Matrices 32
§1.4. The Rank of a Linear Operator and a Matrix 38
§1.5. Determinant and Trace 40
§1.6. Inner Products and Orthonormal Bases 42
§1.7. Orthogonal Direct Sum 49
§1.8. Exercises from the Text 51
§1.9. Additional Exercises 52
Chapter 2. Finite-Dimensional Operator Theory 56
§2.1. Linear Functionals and the Dual Space 56
§2.2. Riesz Representation Theorem and Adjoint Operators 58
§2.3. Self-adjoint and Unitary Operators 61
§2.4. Orthogonal Complements and Projections 67
§2.5. The Moore-Penrose Inverse 74
§2.6. Eigenvalues for Operators 75
§2.7. Square Roots of Positive Operators 82
§2.8. The Polar Decomposition 84
§2.9. Traces of Operators 87
§2.10. The Operator Norm 89
§2.11. The Spectral Theorem 92
§2.12. Exercises from the Text 96
§2.13. Additional Exercises 97
Chapter 3. Introduction to Finite Frames 102
§3.1. R[sup(n)]-Frames 103
§3.2. Parseval Frames 107
§3.3. General Frames and the Canonical Reconstruction Formula 113
§3.4. Frames and Matrices 119
§3.5. Similarity and Unitary Equivalence of Frames 124
§3.6. Frame Potential 128
§3.7. Numerical Algorithms 133
§3.8. Exercises from the Text 136
§3.9. Additional Exercises 136
Chapter 4. Frames in R[sup(2)] 138
§4.1. Diagram Vectors 138
§4.2. Equivalence of Frames 140
§4.3. Unit Tight Frames with Four Vectors 144
§4.4. Unit Tight Frames with k Vectors 146
§4.5. Fundamental Inequality in R[sup(2)] 149
§4.6. Frame Surgery: Removals and Replacements 152
§4.7. Exercises from the Text 153
§4.8. Additional Exercises 154
Chapter 5. The Dilation Property of Frames 156
§5.1. Orthogonal Compressions of Bases 156
§5.2. Dilations of Frames 159
§5.3. Frames and Oblique Projections 164
§5.4. Exercises from the Text 167
§5.5. Additional Exercises 167
Chapter 6. Dual and Orthogonal Frames 170
§6.1. Reconstruction Formula Revisited 170
§6.2. Dual Frames 171
§6.3. Orthogonality of Frames 178
§6.4. Disjoint Frames 183
§6.5. Super-Frames and Multiplexing 187
§6.6. Parseval Dual Frames 189
§6.7. Exercises from the Text 194
§6.8. Additional Exercises 195
Chapter 7. Frame Operator Decompositions 198
§7.1. Continuity of Eigenvalues 199
§7.2. Ellipsoidal Tight Frames 204
§7.3. Frames with a Specified Frame Operator 211
§7.4. Exercises 218
Chapter 8. Harmonic and Group Frames 220
§8.1. Harmonic Frames 220
§8.2. Frame Representations and Group Frames 228
§8.3. Frame Vectors for Unitary Systems 235
§8.4. Exercises 241
Chapter 9. Sampling Theory 244
§9.1. An Instructive Example 244
§9.2. Sampling of Polynomials 246
§9.3. Sampling in Finite-Dimensional Spaces 251
§9.4. An Application: Image Reconstruction 261
§9.5. Sampling in Infinite-Dimensional Spaces 268
§9.6. Exercises 278
Chapter 10. Student Presentations 280
§10.1. Eigenspace Decomposition 280
§10.2. Square Roots of Positive Operators 283
§10.3. Polar Decomposition 285
§10.4. Oblique Projections and Frames 289
§10.5. Vandermonde Determinant 292
Chapter 11. Anecdotes: Frame Theory Projects by Undergraduates 296
Bibliography 302
Index of Symbols 306
Index 308
A 308
B 308
C 308
D 308
E 308
F 308
G 309
H 309
I 309
J 309
M 309
N 309
O 309
P 310
R 310
S 310
T 310
U 310
V 310
W 310
Back Cover 314
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