Wavelets: Basic properties, parameterizations and sampling -- Derivatives and multiwavelets -- Sampling in Fourier and wavelet analysis -- Bases for time-frequency analysis -- Fourier uncertainty principles -- Function spaces and operator theory -- Uncertainty principles in mathematical physics.Developed in this book are several deep connections between time--frequency (Fourier/Gabor) analysis and time--scale (wavelet) analysis, emphasizing the powerful adaptive methods that emerge when separate techniques from each area are properly assembled in a larger context. While researchers at the forefront of developments in time--frequency and time--scale analysis are well aware of the benefits of such a unified approach, there remains a knowledge gap in the larger community of practitioners about the precise strengths and limitations of Fourier/Gabor analysis versus wavelets. This book fills that gap by presenting the interface of time--frequency and time--scale methods as a rich area of work. Topics and Features: * Inclusion of historical, background material such as the pioneering ideas of von Neumann in quantum mechanics and Landau, Slepian, and Pollak in signal analysis * Presentation of self-contained core material on wavelets, sampling reconstruction of bandlimited signals, and local trigonometric and wavelet packet bases * Development of connections based on perspectives that emerged after the wavelet revolution of the 1980s * Integrated approach to the use of Fourier/Gabor methods and wavelet methods * Comprehensive treatment of Fourier uncertainty principles * Explanations at the end of each chapter addressing other major developments and new directions for research Applied mathematicians and engineers in signal/image processing and communication theory will find in the first half of the book an accessible presentation of principal developments in this active field of modern analysis, as well as the mathematical methods underlying real-world applications. Researchers and students in mathematical analysis, signal analysis, and mathematical physics will benefit from the coverage of deep mathematical advances featured in the second part of the work.