Frames for Undergraduates
This book is a gentle introduction to the theory of frames. Any element of a vector space can be reconstructed from its (unique) coefficients in a basis of the space. If, moreover, the basis is orthonormal, the coefficients are given by the scalar products with elements of the basis. It is very convenient in practice; they can be computed very easily and quickly. But to use sets of such coefficients is a vulnerable procedure, e.g. if in the considered process they get incomplete. Elements cannot be reconstructed from an incomplete set of its coefficients. On the other hand, any element of a vector space can be reconstructed from the set of its coefficients with respect to a spanning set. Coefficients are no more unique but often it does not matter. It turns out that in Hilbert spaces there are spanning sets that are not bases but which retain the property that the coefficients of any element of the space are its scalar products with the elements of the spanning set. So they keep the convenient property of easy and quick computation. Such spanning sets are called frames.
The book is a very nice introduction into, and survey of, frame theory. Its value is threefold. Firstly, it provides an introduction into the theory of frames. After a terse summary of the necessary items from linear algebra and operator theory (chapters 1 and 2), the reader is acquainted with the main topics in chapters 3-6. Secondly, chapters 7-9 are written at an advanced level. The reader is motivated by interesting open questions and a wide field of applications. Thirdly, the book is an excellent guide for teachers. It is skilfully written at a high mathematical and pedagogical level. Though its title reads "Frames for Undergraduates", it can be very valuable not just for undergraduates but also for professional mathematicians.