An Introduction to Wavelet Analysis by David F. Walnut (PDF)

Description

This book provides a comprehensive presentation of the conceptual basis of wavelet analysis, including the construction and analysis of wavelet bases. It motivates the central ideas of wavelet theory by offering a detailed exposition of the Haar series, then shows how a more abstract approach allows readers to generalize and improve upon the Haar series. It then presents a number of variations and extensions of Haar construction.

User’s Reviews

Editorial Reviews: Review “[This text] is carefully prepared, well-organized, and covers a large part of the central theory . . . [there are] chapters on biorthogonal wavelets and wavelet packets, topics which are rare in wavelet books. Both are important, and this feature is an extra argument in favour of [this] book . . . the material is accessible [even] to less advanced readers . . . the book is a nice addition to the series.” ―Zentralblatt Math”This book can be recommended to everyone, especially to students looking for a detailed introduction to the subject.” ―Mathematical Reviews”This textbook is an introduction to the mathematical theory of wavelet analysis at the level of advanced calculus. Some applications are described, but the main purpose of the book is to develop―using only tools from a first course in advanced calculus―a solid foundation in wavelet theory. It succeeds admirably. . . . Part I of the book contains 112 pages of preliminary material, consisting of four chapters on ‘Functions and Convergence,’ ‘Fourier Series,’ ‘Fourier Transforms,’ and ‘Signals and Systems.’ . . . This preliminary material is so well written that it could serve as an excellent supplement to a first course in advanced calculus. . . . The heart of the book is Part III: ‘Orthonormal Wavelet bases.’ This material has become the canonical portion of wavelet theory. Walnut does a first-rate job explaining the ideas here. . . . Ample references are supplied to aid the reader. . . . There are exercises at the end of each section, 170 in all, and they seem to be consistent with the level of the text. . . . To cover the whole book would require a year. An excellent one-semester course could be based on a selection of chapters from Parts II, III, and V.” ―SIAM Review”D. Walnut’s lovely book aims at the upper undergraduate level, and so it includes relatively more preliminary material . . . than is typically the case in a graduate text. It goes from Haar systems to multiresolutions, and then the discrete wavelet transform . . . The applications to image compression are wonderful, and the best I have seen in books at this level. I also found the analysis of the best choice of basis, and wavelet packet, especially attractive. The later chapters include MATLAB codes. Highly recommended!” ―Bulletin of the AMS

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐First off, this book assumes at a solid grounding in linear algebra and calculus. That said, if you don’t have a good understanding of these topics I don’t think there is any way to get up to speed on wavelet analysis.The book is well-structured and topics clearly build off of one another. This can be overwhelming at times, as not understanding an earlier theorem might leave you lost as to the significance of a later one. My biggest complaint is that the book is almost entirely composed of proofs and theorems with little motivation. If you’re willing to invest the time to understand the proofs and work out the exercises, the book will leave you with an excellent understanding. If you’re looking for a somewhat gentler introduction with explanations in common English rather than pure math, keep looking.

⭐I take it as a healthy sign when there is a burst of new books in a sub-area of math. In wavelet analysis and its applications, we have seen a number of recent books arrive to university bookstores. Surprisingly there doesn’t in fact seem to be much of an overlap of subject or scope, from one book to the next. The subject is infinite in many directions, for example the kind of student it is aimed at, the level, the specialized area within math itself, and the kind of application it is stressing. D. Walnut’s lovely book aims at the upper undergraduate level, and so it includes relatively more preliminary material, for example Fourier series, than is typically the case in a graduate text. It goes from Haar systems to multirelutions, and then the discrete wavelet transform, starting on page 215. The applications to image compression are wonderful, and the best I have seen in books at this level. I also found the analysis of the best choice of basis, and wavelet packet, especially attractive. The later chapters include MATLAB codes.– Highly recommended!

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