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Frames for Undergraduates

Deguang Han, Keri Kornelson, David Larson, and Eric Weber
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mathematical Library 40
[Reviewed by
William J. Satzer
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Frames for Undergraduates is a volume in the Student Mathematical Library published by the AMS. It was written, at least in part, as a consequence of a Research Experience for Undergraduates (REU) program called “Matrix Analysis and Wavelets”. This book, intended for advanced undergraduates, is an introduction to the theory of frames in a finite-dimensional Hilbert space. Since the concept of a frame is probably not familiar to every reader, let’s take a simple case: an Rn –frame is simply a collection of vectors spanning Rn. (Obvious analogous definitions replace Rn by Cn or any finite-dimensional Hilbert space. The story is different in infinite dimensions, where an alternate definition is necessary.) So an orthonormal basis is a frame, as is any basis. Of course, if that’s all there were to frames, we really wouldn’t need a new word.

However, interesting applications of the idea appear when the collection of vectors in the frame includes “redundant” vectors, more than are needed to span the space. Frames are more general than orthonormal bases, but they can retain some valuable properties of orthonormal bases. For example, certain frames allow one to use inner products to calculate (non-unique) coefficients of a vector in terms of the frame vectors, just as one can calculate (unique) coefficients with orthonormal bases. At the same time, redundancy in a frame means that if a coefficient is lost (as, for example, in a communication system that transmits coefficients to enable reconstruction of a signal at a distant location), the information may be reconstructed from the surviving coefficients. Accordingly, frames can be selected to span the space as uniformly as possible, or the vectors in the frame may be concentrated in certain critical subspaces.

This is motivation, but much of the fun is using the concepts and tools of linear algebra to explore questions raised by the notion of a frame. A Parseval frame is one which satisfies the standard Parseval equality. So an orthonormal basis is a Parseval frame. Are there Parseval frames that are not orthonormal bases? Yes, there are. But many frames are not Parseval frames, and one can characterize frames by providing bounds that show by how much they fail the Parseval equality.

The book has ten chapters. The first two review the necessary background from linear algebra and could be skipped, browsed or used as for reference as needed. (These chapters include material that is not ordinarily part of an introductory linear algebra course such as some finite-dimensional operator theory.) Frames are introduced in Chapters 3; they are defined in finite-dimensional Hilbert spaces, and then studied initially in R2. The authors introduce frame operators and discuss numerical algorithms for reconstructing a vector from its frame coefficients and the frame vectors. Succeeding chapters discuss dilations of frames, as well as dual and orthogonal frames. Chapters 7, 8 and 9 cover more advanced topics. Chapter 9 — unfortunately near the end — discusses sampling theory and digital image reconstruction. This is the real punch line for students looking for applications and trying to understand the “why” of frames. It would be advantageous if at least a taste of this application could be brought forward and discussed earlier in the book.

A final chapter includes material that would be appropriate for students to prepare and present. There are a variety of exercises throughout and a good bibliography.

This book would be a good candidate for a topics course or for a second course in linear algebra. The authors suggest that prerequisites include introductory linear algebra and a semester of proof-based analysis. A strong linear algebra student with experience doing proofs would also find this text accessible.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

  • Introduction
  • Linear algebra review
  • Finite-dimensional operator theory
  • Introduction to finite frames
  • Frames in $\mathbb{R}^2$
  • The dilation property of frames
  • Dual and orthogonal frames
  • Frame operator decompositions
  • Harmonic and group frames
  • Sampling theory
  • Student presentations
  • Anecdotes on frame theory projects by undergraduates
  • Bibliography
  • List of symbols
  • Index