Wavelets: First Steps
Nico Μ. Τemme · WORLD SCIENTIFIC eBooks · 1993
This chapter gives a short elementary introduction to wavelets.We give a few properties of continuous wavelets, a few remarks on multiresolution analysis, and the construction for the first few spline wavelets as solutions of dilation equations.We also describe an example of the compactly supported orthonormal Daubechies wavelets.All this will be discussed in more depth in the following chapters. §1 About the history of waveletsWavelets were introduced at the beginning of the 'eighties by J. Morlet, a French geophysicist at Elf-Aquitane, as a tool for signal analysis in view of applications for the analysis of seismic data.The numerical success of Morlet prompted A. Grossmann to make a more detailed study of the wavelet transform, which resulted in a paper giving the mathematical foundations (see Grossmann & Morlet (7]), where the title of the paper still shows the name wavelets of constant shape.In 1985, the harmonic analyst Y. Meyer became aware of this theory and he recognized many classical results inside it.Meyer pointed out to Grossmann and Morlet that there was a connection between their signal analysis methods and existing, powerful techniques in the mathematical study of singular integral operators.Then Ingrid Daubechies became involved, and all this resulted in the first construction of a special type of frames (see Daubechies, Grossmann & Meyer [3]), (the concept frame generalizes the concept basis in a Hilbert space).It also was the start of a crossfertilization between the signal analysis applications and the purely mathematical aspects of techniques based on dilations and translations.In 1988 Daubechies provided a major breakthrough by constructing families of orthonormal wavelets with compact support (see Daubechies [4]).In this she was inspired by work of Mallat and Meyer in the field of multiresolution analysis, Wavelets: An Elementary Treatment of Theory and ApplicationsTom H.