A few years ago I was chatting with a colleague who was about to retire. He pointed to his book shelf and remarked that he was embarrassed that after thirty-five years most of the books on his shelf were titled An Introduction to… Well, there are introductions, and there are introductions. The book under review, Frames and Bases: An Introductory Course, pushes the notion pretty far. The author says that this text is a streamlined version of his earlier book, Introduction to Frames and Riesz Bases. The current book is, nonetheless, an advanced introduction.
Briefly, a frame is a sequence of elements in a Hilbert space H (possibly finite-dimensional) such that each element of H can be written as a linear combination of the members of the frame. Thus, an ordinary basis is a frame, but the concept is more general. One way to think of it is as a basis with redundant elements. There are significant applications of frames in signal and image processing where, for example, redundancy can help reconstruct a corrupted signal. Applications of frames also appear in image analysis in conjunction with Gabor and wavelet bases.
Of course, frames also enjoy a considerable amount of theoretical interest.
The author’s aim is to present the central ideas of frame and basis theory. He does this in essentially two parts. The first, in Chapters 1 through 5, addresses the fundamental theoretical aspects. Then, in Chapters 7 through 11, he focuses on explicit constructions of frames in L2(R) and subspaces thereof. The linking Chapter 6 introduces B-spline theory and the main properties of B-splines; the author uses these to ensure that the constructions of later chapters are more convenient to apply in practice.
The author suggests that the book is aimed toward graduate students in mathematics, pure and applied mathematicians, and practicing engineers. Required background would be a standard graduate course in real analysis that included the basics of functional analysis. Although the author presents some elementary background material on linear algebra and functional analysis, the treatment is so concise that it really works only as synopsis and review. A reader without a strong background would find this text very challenging.
Bill Satzer (email@example.com) 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.