3 comments about Wavelets: A Primer.

1. This book is a good tool for those who want to explore the mathematical rigors of wavelet theory. The "primer" assumes that the user is a fairly well versed mathematician, but the derivations are complete. There is little to nothing said about the practical applications of wavelets. For applications, look to other books.



2. This booklet, the translation of the original German version "Wavelets - Eine Einfuehrung" (published by Vieweg-Verlag), is indeed a real primer.

   It removes old layers of thinking about the structure of signals and replaces them by an elegant formalism rooted in the theory of Hilbert spaces and orthonormal bases.

   A graduate background on Hilbert spaces (L_2 etc.), the Fourier transform and some integration theory (Lebesgue integration) is recommended.

   There are very few (easily corrected) typos and the book can be used as a basis for a one quarter course of 4 hours each week during 7 weeks (or a semester of 2 hours), with a backing of handout problems on several aspects of the theory.




3. Choosing a book on Wavelets to use as material for an introductory course is far from simple: there are many books on the subject and often the mathematical starting level of the audience is not uniform. But the book by Blatter is in my opinion very good if one wants to start from scratch (well, not quite of course: some knowledge of Lebesgue integration and infinite dimensional Banach and Hilbert spaces of functions is needed) in a mathematical or technical environment.

   Starting with 2 chapters on Fourier analysis, going from windowed Fourier transform to a first introduction of continuous wavelet transform (CWT), and giving connections to several well known facts from signal analysis (the Heisenberg uncertainty priciple and the Shannon sampling theorem), the theoretical framework of the CWT is given in chapter 3. Interesting are the results on the decay of the wavelet transform in terms of the smoothness of the signal and the number of vanishing moments for the wavelet.

   This is followed by a chapter on the important subject of 'frames' and one on multiresolution analysis (the foundation for the hierarchy of orthonormal bases for L^2(R)). The scaling function (time domain) is actuallly constructed from its properties in the Fourier domain (i.e. the frequency domain).

   The final chapter treats the basic idea behind and constructions of orthonormal wavelets with compact support (including the Daubechies wavelets), also touching upon binary interpolation and so called spline wavelets.

   For applications to a host of disciplines it is possible to consult one of the many applied books: in my opinion it is necessary to have a firm grasp of the underlying theoretical framework before really understanding what one is doing. I have nothing against applications, but once one has to go back to the roots: back to the valley to be able to reach the crest of the wavelet without accidents.

No comments about Wave Packet Analysis (Cbms Regional Conference Series in Mathematics) (Cbms Regional Conference Series in Mathematics).

1 comments about The General Theory of Dirichlet's Series (Phoenix Edition).

1. This is a short (68 pages),intense and excelent book where are studied the Dirichlet's series ( the results are former to 1915). The author is an eminent mathematician, a great number theorist, G. H. Hardy.
   
   Rafael Jakimczuk

No comments about Mathematical Properties of Sequences and Other Combinatorial Structures (The Springer International Series in Engineering and Computer Science).

1 comments about Wavelets and Multiwavelets (Studies in Advanced Mathematics).

1. While there has been a number of recent wavelet books on the market covering a variety of aspects of this important subject of applied math; of those books only a few are good for use in a beginning course; the kind of course aimed at advanced undergraduates, and at students in related fields of math such as numerical analysis; and students from other departments, especially students from engineering departments: EE, medical imaging, and Computer and Electrical Engineering; to mention just a few.
   The present book seems especially pedagogical well designed for student needs; and the author clearly had students in mind. Rather than delving into a variety of technical issues, the book goes directly to the underlying geometric ideas that students can understand at an intuitive level; without too much background. The algorithmic steps in the subject well presented, and they are essential both for the theory, and for the applications to the analysis of number sets (data mining), of pictures, and of many other applications.
   The book is certainly more limited in scope than for example Daubechies's book that appeared in the SIAM series in 1992, but it is also gentle on students, and is likely to serve as a lovely invitation to the subject.
   When the students progress in the course, they will then build up a deeper understanding of the fundamental ideas, which in fact are quite simple when isolated from the traditional rather heavy notation and terminology. The fundamental ideas are nicely illustrated with tree diagrams which also serve as the underpinning for wavelet programs.
   An additional advantage of this little book is that students will be able to get hands-on experience with graphics, and with MATLAB routines, as they work their way through the book.
   Review by Palle Jorgensen, December 2004.

No comments about Harmonic Analysis, the Trace Formula, and Shimura Varieties: Proceedings of the Clay Mathematics Institute, 2003 Summer School, the Fields Institute, (Clay Mathematics Proceedings,).

1 comments about A First Course on Wavelets (Studies in Advanced Mathematics).

1. The Hernandez-Weiss book is a great source in mathematics, where the reader can learn about
   some fundamental and timeless tools of wavelet analysis. The presentation is clear and self contained. The book is great for an upper level analysis course, as well as for self study. Its emphasis is centered around the
   multiresolution method, and it has both the basics in the subject, and
   some exciting new results,-- those due
   to the co-authors, as well as those of the analysts in the extended "Wash U-family" of researchers. It has a wealth of new material that
   came after the 1992 classic of I. Daubechies ("Ten lectures on wavelets" SIAM ).The presentation in Hernandez-Weiss of, for example,
   multiplicity theory and the dimension function is great for
   the student who wishes to pick up the central ideas, or to
   learn the details, step by step. The book contains both history and
   motivation, and it points the reader towards a variety of exciting and timeless applications.

No comments about Essays in Commutative Harmonic Analysis (Applied Mathematical Sciences).

No comments about Sequences, Discrepancies and Applications (Lecture Notes in Mathematics).

3 comments about Introduction to Fourier Analysis and Wavelets (Brooks/Cole Series in Advanced Mathematics).

1. Courses in harmonic analysis have a central place in the course offerings of every math department, be it pure or applied;-- and the subject is as important as ever! Yet it has not always been easy for an instructor to find a book that is right for the students. Some books might be too skimpy on proofs, or not deep enough.-- Or the applications may somehow be artificial, or contrived. Afterall, we teach the material to engineers!-- It is a relief to find, in Pinsky's lovely new book, a balanced approach to the subject. The motivation and the history receive a beautiful presentation, as do the technical points and proofs. And the historical comments- sprinkled throughout the book- bring the subject to life. At the same time, the book is forward looking, and it has been tested in courses. Great exercises! The structure of the exposition is friendly, and gently leads the reader toward the exciting new wavelet material in the last hundred or so pages of the book. The student thereby gets a sense of how the central questions in wavelet theory have their root in the more classical ideas of harmonic analysis.



2. As a probabilist and statistician, with a Ph.D., having worked at universities, as a consultant, and in industry for approximately forty years, I had previously employed Fourier Analysis only as a tool, not having studied the subject as a discipline unto itself. Dr. Pinsky's book has allowed me to learn the subject more deeply, and from a different, exciting viewpoint. In addition, I needed a resource that would permit me to learn about wavelets. How wonderful to find a book that includes both topics! Moreover, this book is a pleasure to read, with pencil and paper, to work through the ideas. It is extraordinarily well-written, which is not surprising, given the clarity and excellence of Dr. Mark A. Pinsky's other works. Pinsky's grammar is excellent, which is extremely refreshing. Many modern authors cause me to believe that I should have a red pen to correct grammatical errors while reading their works. This book is much more than just a textbook; it is beautiful mathematically and beautifully written.



3. I needed to learn Fourier Analysis and Wavelets, and this book is excellent as a textbook and as a reference. It is also quite readable. We need more mathematics books like this one.