Wavelets: A Primer Wavelets. English

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. Contents......Page 3 Preface......Page 5 Read Me......Page 7 1.1 A central theme of analysis......Page 9 1.2 Fourier series......Page 12 1.3 Fourier transform......Page 16 1.4 Windowed Fourier transform......Page 19 1.5 Wavelet transform......Page 22 1.6 The Haar wavelet......Page 28 2.1 Fourier series......Page 37 2.2 Fourier transform on R......Page 42 2.3 The Heisenberg uncertainty principle......Page 57 2.4 The Shannon sampling theorem......Page 61 3.1 Definitions and examples......Page 69 3.2 A Plancherel formula......Page 77 3.3 Inversion formulas......Page 82 3.4 The kernel function......Page 86 3.5 Decay of the wavelet transform......Page 90 4.1 Geometrical considerations......Page 98 4.2 The general notion of a frame......Page 107 4.3 The discrete wavelet transform......Page 112 4.4 Proof of theorem (4.10)......Page 122 5 Multiresolution analysis......Page 128 5.1 Axiomatic description......Page 129 5.2 The scaling function......Page 134 5.3 Constructions in the Fourier domain......Page 142 5.4 Algorithms......Page 157 6.1 The basic idea......Page 165 6.2 Algebraic constructions......Page 176 6.3 Binary interpolation......Page 184 6.4 Spline wavelets......Page 196 References......Page 207

The Wavelet Transform has stimulated research that is unparalleled since the invention of the Fast Fourier Transform and has opened new avenues of applications in signal processing, image compression, radiology, cardiology, and many other areas. This book grew out of a short course for mathematics students at the ETH in Zurich; it provides a solid mathematical foundation for the broad range of applications enjoyed by the wavelet transform. Numerous illustrations and fully worked out examples enhance the book.

Booknews

An introduction to the new mathematical technique of wavelet transform, which has been applied to the fields of signal processing, image compression, radiology, and cardiology. Features multiresolution analysis with fast algorithms and the construction of orthonormal wavelets with compact support. Originally published in German by Vieweg-Verlag. Annotation c. by Book News, Inc., Portland, Or.

This Book Grew Out Of A Short Course For Mathematics Students At The Eth In Zurich; It Provides A Solid, Yet Accessible, Mathematical Foundation For Those Interested In Learning About Wavelets And Pursuing The Broad Range Of Applications For Which The Wavelet Transform Has Proved Successful. Numerous Illustrations And Fully Worked-out Examples Further Enhance The Value Of This Exemplary Introduction To The Field.--jacket. 1. Formulating The Problem -- 2. Fourier Analysis -- 3. The Continuous Wavelet Transform -- 4. Frames -- 5. Multiresolution Analysis -- 6. Orthonormal Wavelets With Compact Support. Christian Blatter. "This book grew out of a short course for mathematics students at the ETH in Zurich; it provides a solid, yet accessible, mathematical foundation for those interested in learning about wavelets and pursuing the broad range of applications for which the wavelet transform has proved successful. Numerous illustrations and fully worked-out examples further enhance the value of this exemplary introduction to the field."--BOOK JACKET. A foundation text for those interested in learning about wavelets and pursuing the broad range of applications for which the wavelet transform has proved successful, including signal processing, image compression, radiology and cardiology. The approximation, resp. the representation, of arbitrary known or unknown functions f by means of special functions can be viewed as a central theme of analysis.