Wavelets: Basic properties, parameterizations and sampling

Abstract

Wavelets play a fundamental role in decomposing the function spaces—and operators that act on them—that are considered throughout this book. Morlet and colleagues (e.g., [278, 279]) coined the term ondelettes to describe families of shifted and modulated pulses generated from a single function ψ. Their discovery of the benefits of wavelets in geoexploration was quickly seen as a germination of similar ideas incubating collectively in the mathematics (e.g., Calderón-Zygmund theory), mathematical physics (e.g., coherent states) and electrical engineering (vis-à-vis subband coding) communities. The key features of wavelets that enable their exploitation in discrete signal analysis have long since been distilled into the conceptual framework of a multiresolution analysis (MRA).