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As Y. Meyer wrote in [M2]: äAt the beginning of the 1980's, many scientists were already using ‘wavelets’ as an alternative to traditional Fourier analysis. This alternative gave grounds for hoping for simpler numerical analysis and more robust synthesis of certain transitory phenomena.” He also wrote: “To mention only the most striking, R. Coifman and G. Weiss invented the ‘atoms’ and ‘molecules’ which were to form the basic building blocks of various function spaces, the rules of assembly being clearly defined and easy to use. Certain of these atomic decompositions could, moreover, be obtained by making a discrete version of a well-known identity, due to A. Calderón, in which ‘wavelets’ were implicitly involved. That identity was later rediscovered by Morlet and his collaborators.” Y. Meyer further wrote: “These separate investigations had such a ‘family resemblance’ that it seemed necessary to gather them together into a coherent theory, mathematically well-founded and, at the same time, universally applicable.”
Today we know that this coherent theory is wavelet analysis. This theory played and will, doubtless, play an important role in many different branches of science and technology. Wavelet analysis provides a simpler and more efficient way to analyze those functions and distributions that have been studied by use of Fourier series and integrals. But, however, Fourier analysis still plays a key role in constructing the orthonormal bases of wavelets.
KeywordsWavelet Analysis Homogeneous Space Besov Space Functional Space Convolution Operator
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