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Frames for undergraduates. (English) Zbl 1143.42001
Student Mathematical Library 40. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4212-6/pbk). xiv, 295 p. (2007).
Around the turn of the millennium it became a popular pedagogical practice, at least in the US, to introduce every abstract concept in undergraduate mathematics first through an application. There seems to be a backlash to this approach now. Linear algebra is one of those interesting topics in undergraduate mathematics that allows almost effortless interplay between abstract theory and applications, but too many diverse applications can obstruct even a very bright undergraduate from obtaining a sense of the structure of the theory of linear mappings. This book on frames can be regarded as a second course in linear algebra, focusing first on the abstract theory – both as a review and as a bridge to more advanced techniques needed for what is to come – and only later on more sophisticated applications. The theory of frames, as introduced here, provides a natural context for helping a student make conceptual connections between algebraic and analytic properties of linear mappings and linear spaces, as well as more advanced geometric perspectives. Since everything is done in finite dimensions, none of the subtleties attached to questions of convergence get in the way.
This little book is based on a multiyear summer research experience for undergraduates (REU) program at Texas A&M University. The theory of frames emerged from the study of nonharmonic Fourier series in the work of R.J.Duffin and A.C.Schaeffer [Trans. Am. Math. Soc. 72, 341–366 (1952; Zbl 0049.32401)] and, more recently, in Gabor theory and wavelet theory. A frame \(\mathcal{F}\) is a complete family \(\{f_n\}_n\) of functions in a (separable) Hilbert space that has an important stability property, namely, that the coefficient mapping \(T:f\mapsto \sum_n \langle f,\, f_n\rangle\) and the synthesis mapping \(T^\ast: \{c_n\}\mapsto \sum_n c_n f_n\) give rise to a composition \(T^\ast T\) that is bounded and continuously invertible on \(H\). This amounts to the existence of constants \(A\) and \(B\) such that, for all \(f\in H\), one has \[ A\|f\|^2\leq \sum_n |\langle f,\, f_n\rangle|^2 \leq B\|f\|^2. \] The supremum of \(A\) and infimum of \(B\) such that these inequalities hold are called the frame bounds. The frame \(\mathcal{F}\) is said to be tight if one can take \(A=B\). These concepts are advanced when one is talking about infinite-dimensional Hilbert spaces, but they are relatively straightforward extensions of bases in finite-dimensional settings. Restricting to finite dimensions for the sake of making the concept accessible at an elementary level seems artificial until one is made aware of the historical development of the theory of frames. Some of the deeper and certainly more useful ideas in frame theory (re)emerged in the context of finite-dimensional spaces. Thus, bringing an undergraduate up to speed in frame theory amounts in some respects to bringing that student to the forefront of applied analysis.
One visualizable concept in frame theory is that of frame potential \[ P_{\mathcal F} = \sum_{n,m} |\langle f_n,\, f_m\rangle| \] introduced by J.J.Benedetto and M.Fickus [Adv. Comput. Math. 18, No. 2–4, 357–385 (2003; Zbl 1028.42022)]. In the case of normalized frames (i.e., \(\|f_n\|=1\) for all \(n\)) with \(K\) elements in an \(N\)-dimensional Hilbert space, \(P_{\mathcal{F}}\) is minimized over all such frames precisely when the frame is tight.
Tight frames are generalizations of orthonormal bases in the sense that, for a tight frame \(\mathcal{F}\) and \(f\in H\), one has \[ f={1\over A}\sum_n \langle f,\, f_n \rangle f_n. \] When this formula applies for all \(f\) and with \(A=1\), the frame \(\{f_n\}\) is called a Parseval frame since it admits the same formal expansion as does an orthonormal basis. In an \(N\)-dimensional Hilbert space, any frame with \(K>N\) elements cannot be linearly independent so it is interesting to ask: what is the relationship between Parseval frames and orthonormal bases. It turns out that any \(K\)-term Parseval frame is the image of an orthonormal basis in a \(K\)-dimensional Hilbert space under a projection operator. When working in coordinates and regarding the frame element \(f_n\) as a column vector in \(\mathbb{R}^n\), the matrix \(M=[f_1,\dots f_K]\) can be dilated to a unitary matrix \(U=[M^T, N^T]^T\), where \(N\) is a \((K-N)\times K\) matrix. This fact turns out to be a special case of Naimark’s dilation theorem, cf. e.g., W.Czaja [Proc. Am. Math. Soc. 136, No. 3, 867–871 (2008; Zbl 1136.42027)], though an elementary proof was given by D.-G.Han and D.R.Larson [“Frames, bases and group representations”, Mem. Am. Math. Soc. 697 (2000; Zbl 0971.42023)]. Naimark’s result is deep, beautiful and far reaching, but only accessible to a small population. The elementary approach here is accessible to anyone with a decent understanding of linear algebra.
After discussing this basic theory, the authors go on to develop further basic theory of dual frames plus other analytic properties such as continuity of eigenvalues and ellipsoidal tight frames. The notion of a dual frame is very useful in contexts in which, for a given frame, a choice of dual (analyzing functions) that minimize some sort of cost functional can be considered. The perspective gained from Naimark’s theorem makes the study of dual frames less of a mystery.
Harmonic or Fourier frames involving samples of complex exponentials provide a good illustration of several aspects of the theory developed to this point, as well as a transition to frame theory in the context of unitary systems (such as the Gabor and wavelet systems in which the theory of frames came to fruition). The final chapter involves applications to image processing.
In summary, 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. Used in one of these ways, a good undergraduate might achieve a sophisticated understanding both of the linear geometric aspects of finite-dimensional functional analysis and of some of current applications in signal and image processing.

MSC:
42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
42C15 General harmonic expansions, frames
47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
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