A First Course on Wavelets (Studies in Advanced Mathematics) 1st Edition by Eugenio Hernandez (PDF)

Description

Wavelet theory had its origin in quantum field theory, signal analysis, and function space theory. In these areas wavelet-like algorithms replace the classical Fourier-type expansion of a function. This unique new book is an excellent introduction to the basic properties of wavelets, from background math to powerful applications. The authors provide elementary methods for constructing wavelets, and illustrate several new classes of wavelets. The text begins with a description of local sine and cosine bases that have been shown to be very effective in applications. Very little mathematical background is needed to follow this material. A complete treatment of band-limited wavelets follows. These are characterized by some elementary equations, allowing the authors to introduce many new wavelets. Next, the idea of multiresolution analysis (MRA) is developed, and the authors include simplified presentations of previous studies, particularly for compactly supported wavelets.Some of the topics treated include:Several bases generated by a single function via translations and dilations Multiresolution analysis, compactly supported wavelets, and spline wavelets Band-limited wavelets Unconditionality of wavelet bases Characterizations of many of the principal objects in the theory of wavelets, such as low-pass filters and scaling functionsThe authors also present the basic philosophy that all orthonormal wavelets are completely characterized by two simple equations, and that most properties and constructions of wavelets can be developed using these two equations. Material related to applications is provided, and constructions of splines wavelets are presented. Mathematicians, engineers, physicists, and anyone with a mathematical background will find this to be an important text for furthering their studies on wavelets.

User’s Reviews

Editorial Reviews: From the Back Cover This unique book is an excellent introduction to the basic properties of wavelets. The fundamental construction of these functions by means of “multiresolution analyses” is presented; in particular, this method is used for introducing the spline wavelets and the compactly supported wavelets. An important feature of this book, however, is the use of the Fourier transform for studying wavelets on the real line. A simple characterization of all wavelets is presented which is most useful for the construction of new families of wavelets. This technique can also be used for obtaining characterizations of low pass filters and scaling functions. Another feature is the use of wavelets for representing those function spaces that are most often encountered in analysis: the Lebesgue spaces, Hardy spaces, and more generally, the Besov spaces, the Sobolev, and the Lipschitz spaces. Other topics, some related to applications, are also included: the Fast Fourier Transform, wavelet packets, frames, local cosine and sine bases and their discrete versions are just some examples.

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

⭐The Hernandez-Weiss book is a great source in mathematics, where the reader can learn aboutsome 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 themultiresolution method, and it has both the basics in the subject, andsome exciting new results,– those dueto 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 thatcame 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 forthe student who wishes to pick up the central ideas, or tolearn the details, step by step. The book contains both history andmotivation, and it points the reader towards a variety of exciting and timeless applications.

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