The textbook is a reprint of the original from 1994. At that time, wavelet analysis just started its overwhelming drive as a tool for signal and image processing. Taking into account the importance of wavelet theory in a series of applications in physics and engineering, there was a need for writing a textbook that is more suitable for students and researchers with a limited knowledge of the functional analytic background. The book by Gerald Kaiser aims at this goal. While a lot of new textbooks on wavelet analysis and its applications have been published, the “Friendly Guide to Wavelets” is still an interesting piece of literature, taking a very special view to the tools of Fourier and wavelet analysis for applications in signal processing. Especially concerning part II on physical wavelets, this book is very unique.
Let me give a short description of the chapters of this book. The first eight chapters are suitable for a graduate course on wavelets for students in mathematics, physics or engineering. Each chapter ends with some suitable exercises. The introduced tools are well motivated by examples. Chapter 1 summarizes some preliminaries from linear algebra, function spaces and the Fourier transform, including the Plancherel theorem and the Parseval identities. Chapter 2 focuses on the windowed Fourier transform that can be seen as a forerunner of the continuous wavelet transform. Chapter 3 contains the continuous wavelet transform. The inversion formula is derived and the time-frequency localization of functions is discussed. In Chapter 4, the theory of generalized frames is studied as a general method for the analysis and reconstruction of signals. Generalizing the Shannon sampling theorem, the author develops a discrete time-frequency analysis in Chapter 5, and a new reconstruction formula is shown that uses only a discrete subset of samples. Chapter 6 aims at a direct discretization of the continuous wavelet transform considered in Chapter 3. The salient feature of discrete wavelet analysis is that the sampling rate is automatically adjusted to the scale. A main tool to construct discrete wavelet bases is the multiresolution analysis presented in Chapter 7. Here, one also finds the description of wavelet filter banks of perfect reconstruction (called subband filtering). Chapter 8 is devoted to Daubechies’ orthonormal wavelet bases that play a special role in signal processing because of their compact support that leads to FIR wavelet filter banks.
Part II of the book considers some special applications of wavelet analysis in electromagnetics, scattering theory and acoustics. In Chapter 9, the wavelet ideas are applied to solutions of the Maxwell equation. The obtained electromagnetic wavelets can be employed for radar signal analysis and electromagnetic scattering in Chapter 10. Finally, the ideas of wavelet theory are also applied to acoustic waves.