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

1. The textbook, actually a collection of lectures, provides a lot of material and examples, and it tries to lead the reader from the basics of subject to the applications of wavelets. Despite of the wide range presented by the chapters, the high reciprocal independence and the completness of proposed path, as an introductory book I would prefer the classical "Ten Lectures on Wavelets", even if Benedetto-Frazier work could be more suitable to get at-a-glance what people are doing (i.e. were doing at the moment of publication) with wavelets.

2 comments about The Mathematics of Juggling.

1. Polster manages to take the innocuous pastime of juggling and segue from it into number theory. The movement of balls is shown to map naturally into the concepts of finite state graphs. Bringing in ideas of permutations. And introducing juggling matrices!
   
   Plus, he points out that the founder of Information Theory, Claude Shannon, was also interested in the theory of juggling. Several crucial juggling theorems were discovered by Shannon and are named after him. For readers familiar with Shannon in computing, the book gives a look at relatively little known research by him.



2. I am a mathematical dunce. I have a better than average IQ, I passed high school algebra, I have been juggling for 35 years, and I even came up with diagrams on my own to remember and devise patterns. I thought I would enjoy this book, and I imagine I would if it were at all comprehensible to me. Though the book is hyped as "useful," "accessible" and "entertaining," buyers should be warned that Polster's book is about math and is written entirely in the language of math. There is no effort to bridge the gap between a practical understanding of juggling logic and his numeric abstractions. He writes in plain English prose until 3/4 of the way down page 8, and by the bottom of that page he has ditched you and disappeared into a world of opaque notation that might as well have been written by G.E. Moore and Bertrand Russell. If page after page of greek letters, academic jargon and abstract equations is easy reading for you, get this book. If you don't already speak higher math, Polster isn't going to teach you how.
   
   For a book that does what Polster does not, find Laws of Form by G. Spencer Brown.

No comments about Fourier Analysis (Graduate Studies in Mathematics) (Graduate Studies in Mathematics).

No comments about An Introduction To The Theory Of Infinite Series.

5 comments about A Friendly Guide to Wavelets.

1. I bought this book when I was particularly interested in continuous wavelet transforms. I found myself flying through the exposition. I felt confident about the knowledge I was acquiring and I was quickly able to apply it. Although I come at this book as a mathematician, I think that it is ideal for engineers and physical scientists who usually have far better grounding in signal processing and related issues in Fourier analysis than do mathematicians. I have recommended this book to students, friends and colleagues with high praise.



2. It is not a friendly guide to Wavelets. However, chapters 1 one to six are excelent. Chapter 7, Multiresolution Analysis, the heart of the matter is hard to understand. So, you get the flavor of Wavelets but not learn to use it.



3. I really did find this lovely book reader friendly! The author has succeeded in communicating the exciting subject of wavelets, and their many applications, to students and more advanced readers alike. He realized that different communities, math, computer science, engineering, and physics,--
   that they have variations in their emphasis, their terminology,
   and their thinking. When authors speak to the various user groups, and do it well,-- like in this case, the result is a cucess. This friendly book is!



4. Some reviewers have commented that this is not such a "friendly"
   guide if you are not friends with upper division or graduate
   level mathematics. There is some truth to this. Unlike the
   pragmatic approach taken in "Ripples in Mathematics" this is
   a mathematical coverage of wavelets.

   For me the value in this book is that it provides a clear
   introduction to the notation and theory behind wavelets. This
   book provides the tools you need to understand the wavelet
   literature better. If you are a software engineer searching
   for a quick guide to wavelet algorithms, this book may disappoint.




5. A classic text *but* look at the graph on the cover very carefully because it is the only one you'll get until you reach the last chapters of the book. In terms of the presentation of the basic theorems and equations, the text is excellent ... BUT ... there is precious little to guide a student to an intuitive and practical understanding of the theory. In practice, one looks at the graphical representations of wavelets quite often -- just like the ones used for Fourier analysis (can you imagine your first Fourier analysis class with no graphs of convolution, impulses, etc? Wavelets *are* Fourier analysis taken a step further in order to handle time varying systems. Graphs are essential to the uninitiated.) In fact, it is helpful to think of wavelets (i.e., wavelet packet decompositions) as a Fourier spectrum with an additional time axis added so you can see how the spectrum changes over time. The "wavelet" method simply optimizes the resolution used at different frequencies. True, it is a little more complicated than that, but it is easy to lose sight of the simple elegance of the topic when so many equations are flung at you without intuitive context. The topic could be presented MUCH better, so don't worry that it takes a long time to get through this material. Read other basic wavelet texts first and then *definitely* come back to this one. You will appreciate its otherwise *excellent* presentation much more and the principles will be easier to put into actual practice rather than mere academic conjecture.

3 comments about An Introduction to Harmonic Analysis.

1. Katznelson's book considers harmonic analysis primarily on the circle group. He does this from a thorougly modern point of view. An understanding of the basic ideas of Banach spaces is required. This book should be on the shelf of any aspiring Harmonic Analyst, especially one with an abstract viewpoint.



2. This is a great book for looking at classical harmonic analysis: the study of Fourier Series on the "typical" groups, includes a quick look at the general situation and ends with an introduction to commutative Banach Algebras. Both topics are continued in [Loomis].



3. When the first edition of Katznelson's book appeared back in 1968 (when I was a student), it soon became the talked about, and universally used, reference volume for the standard tools of harmonic analysis: Fourier series, Fourier transforms, Fourier analysis/synthesis, the math of time-frequency filtering, causality ideas, H^p-spaces, and the various incarnations of Norbert Wiener's ideas on the Fourier transform in the complex domain, Paley-Wiener, spectral theory, and more. It is easy to pick up the essentials in this lovely book. Now, many years later, I occasionaly ask beginning students what their favorite reference is on things like that, and more often than not, it is Katznelson. Thanks to Dover, it is on the shelf of most university bookstores, and priced under US$ 10.

1 comments about Introduction to Fourier Analysis on Euclidean Spaces. (PMS-32).

1. This book deals with the extension of real and complex methods in harmonic analysis to the many-dimensional case. So, its pre-requisites are a strong background in real and complex analysis and some acquaintance with elementary harmonic analysis, that is, this book is intended for graduate students and working mathematicians. Maybe some advanced undergraduates could cover certain parts of the material.

   This book is one component of the Stein trilogy on harmonic analysis (together with "Singular Integrals and Differentiability Properties of Functions" and "Harmonic Analysis", both also reviewed by myself), and as such it must be regarded as an authoritative reference on the subject since Elias Stein and Guido Weiss are two of the leading experts in the field, and the material they selected was taken from their teaching and research experience.

   The contents of the book are: The Fourier Transform; Boundary Values of Harmonic Functions; The Theory of H^p Spaces on Tubes; Symmetry Properties of the Fourier Transform; Interpolation of Operators; Singular Integrals and Systems of Conjugate Harmonic Functions; Multiple Fourier Series.

   Includes motivation and full explanations for each topic, excercises for each chapter, called "further results", and extensive references. Outstanding printing quality and nice clothbound.

   These three volumes should be present in every analyst's library.

   Please take a look to the rest of my reviews (just click on my name above).

1 comments about Sequences, Combinations, Limits (Library of School Mathematics, V. 3).

1. This text consists of three chapters of problems, one each on sequences, combinatorics, and limits of sequences. It was designed to supplement mathematics instruction for students in grades 9 and 10, the last two years, of the Soviet school system. The focus is on proving theorems and many of the problems, which are well chosen, are quite challenging. There are hints in the back of the book and complete solutions are provided. Reading these solutions can be instructive because of the insights provided by the authors. There is also a brief section of test problems for which no solutions are provided.
   
   The first chapter, written by S. I. Gelfand, contains a brief exposition on sequences and mathematical induction. This exposition is followed by problems on mathematical induction, sequences of differences, and arithmetic and geometric progressions. The problem solving techniques introduced in this chapter prove useful in the remainder of the text.
   
   The second chapter, written by M. L. Gerver and A. G. Kushnirenko, consists of problems on combinatorics including combinations, the Binomial Theorem, and applications to algebra, number theory, and geometry. American readers may not be familiar with the notation C_{n}^{k}, which means the number of ways k objects can be chosen from n objects when order does not matter (combinations). The problems in this section are the most tractable and the authors suggest that the reader may want to begin with this section.
   
   The final chapter, written by A. A. Kirillov, consists of problems on limits of sequences (including one on a periodic continued fraction) and series, the Triangle Inequality, and the use of quantifiers. Many of these problems would be suitable for a college course in real analysis.
   
   The final author, N. N. Konstantinov, helped select problems and contributed to the editing of the text.
   
   Since this is a problem book and there is little in the way of exposition, product and summation notation are not introduced and very little terminology is defined. Consequently, the reader would benefit from working through these problems while reading an accompanying expository work on these topics.

No comments about Symmetries and Laplacians: Introduction to Harmonic Analysis, Group Representations and Applications.

3 comments about Introduction to Analysis of the Infinite: Book II.

1. John D. Blanton did a good job in translating Euler's work on analysis of the infinite. Although I did not read the original copy the book, I could still receive the charm of Euler from the translation. Euler wrote this book because he realized that many mathematics students were lacking of the knowledge of analysis of the infinite. He, as a great master of mathematics and educator and with all his passion, presented this timeless masterpiece to all of us. I recommend this book to every students who want to excel in mathematics!



2. The title of the translation is wrong. Anyone who wrote (or writes) in Latin is well aware that the ending -orum is genitive plural, not singular. Clearly, therefore, Euler did mean his book title to read 'Introduction to the Analysis of Infinities.' In effect, the translator says that he changed this, because it doesn't accord with modern mathematics. That is, the plural term 'infinities' is archaic. All current mathematicians (who have looked into the matter) accept that not all infinities are the same size. The implication of this is that there exists more than one 'infinity' (if we're going to talk about 'infinity').
   Mathematics are similar in all languages, so the book is still decent (though not as good as it could be).



3. This is a seminal text by one of history's greatest mathematicians. Unique to his great mathematical peers, Euler was also an extraordinary teacher and expositor. His enthusiasm and genius pour through the pages of this book, with Euler making his characteristically bold and ingenious symbolic arguments to come up with many of the well known formulas that were probably mentioned in your math class. For example, Euler brilliantly uses basic algebra (plus infinitesimals) to come up with some very deep and beautiful formulas, such as Sine's infinite product, e's continued fraction expansion and much more. In fact, if you have ever wondered how all of Euler's beautiful formulas that you saw in class were actually derived; here is your chance to get it straight from the genius who discovered them!
   
   As with any book by a mathematician of the highest rank, this is wholly different from any modern "textbook" and should NOT be considered as such. The should be used for self study or as a compliment to a calculus course, or perhaps most of all (like it was intended in those days believe it or not), be read for the pure enjoyment of the subject. Its format is much more flowing and intuitive than a modern textbook; Euler presents clearly stated mathematical arguments (numbered in order), which he then uses and cites later on to produce more mathematical arguments. He also seems to subtly encourage the reader to pursue various ideas for themselves, lending a certain adventurous quality that in NEVER encountered in the "modern" crap texts.
   
   Be forewarned though; it is NOT for the symbolically weak. If you lack skills in basic algebra its best to brush up before you read this book. Just because it is a "pre-calculus" text does not at all mean that this is elementary. This IS however a relatively easy read IMO due to Euler's intuitive style. Euler is the by far the most accessible compared to his modern peers; Newton and Gauss.