This book, combining analysis and tools from mathematical probability, focuses on a systematic and novel presentation of recent trends in pure and applied mathematics: the emergence of three fields, wavelets, signals and fractals. The unity of basis constructions and their expansions is emphasized as the starting point for the development of bases that are computationally efficient for use in several areas from wavelets to fractals.The book brings together tools from engineering and math, especially from signal- and image processing, and from harmonic analysis and operator theory. The presentation is aimed at graduate students, as well as users from a diverse spectrum of applications.