Wavelets, Multiresolution Analysis and Fast Numerical Algorithms

Abstract Notwithstanding a short period of time since the word “wavelets” was first used to designate classes of functions which form bases of functional spaces, a set of ideas associated with the wavelet transform is already having a significant impact in science and engineering. To be sure, many of the ideas behind the wavelet transform have appeared independently in different fields of mathematics, electrical engineering, physics and computer science. These ideas arose to ad dress the limitations of the Fourier transform as a tool for analyzing signals and images in applications, and functions and operators in mathematics and physics. For example in image processing (Burt and Adleson 1983) and in seismics (Goupillaud, Grossman and Mor let 1984), multiresolution methods were developed in a search for a substitute for signal processing algorithms based on the Fourier transform. The technique of subband coding using quadrature mirror filters ( QMF) with the “exact reproduction property” was introduced in Smith and Barnwell (1986). Coherent states were studied in quantum mechanics (see references in Klauder and Skargerstam 1985, Littlewood and Paley 1937) while Calder6n-Zygmund theories (Calderon and Zygmund 1957) were developed in mathematics. For a historical account see Meyer (1990). In numerical analysis the fast multipole method (FMM) for computing potential interactions was constructed (Greengard and Rokhlin 1987, Carrier, Greengard and Rokhlin 1988). But it is the introduction of wavelets and the notion of multiresolution analysis that allows us to develop a unified perspective on these developments.