Wavelets, multiresolution analysis and fast numerical algorithms

These lectures are devoted to fast numerical algorithms and their relation to a number of important results of Harmonic Analysis. The representation of wide classes of operators in wavelet bases, for example, Calderón-Zygmund or pseudo-differential operators, may be viewed as a method for their “compression”, i.e., conversion to a sparse form. The sparsity of these representations is a consequence of the localization of wavelets in both, space and wave number domains. In addition, the multiresolution structure of the wavelet expansions brings about an efficient organization of transformations on a given scale and interactions between different neighbouring scales. Such an organization of both linear and non-linear transformations has been a powerful tool in Harmonic Analysis (usually referred to as Littlewood-Paley and Calderón-Zygmund theories, see e.g. [33]) and now appears to be an equally powerful tool in Numerical Analysis. In applications, for example in image processing [12] and in seismics [22], the multiresolution methods were developed in a search for a substitute for the signal processing algorithms based on the Fourier transform. A technique of subband coding with the exact quadrature mirror filters (QMF) was introduced in [37]. It is clear that the stumbling block on the road to both, the simpler analysis and the fast algorithms, was a limited variety of the orthonormal bases of functional spaces. In fact, there were (with some qualifications) only two major choices, the Fourier basis and the Haar basis. These two bases are almost the antipodes in terms of their space–wave number (or time-frequency) localization. Therefore, it is a remarkable discovery that besides the Fourier and Haar bases, there is an infinite number of various orthonormal bases with a controllable localization in the space–wave number domain. The efforts in mathematics and various applied fields culminated in the development of orthonormal bases of wavelets [39], [30], and the notion of Multiresolution Analysis [31], [27]. There are many new constructions of orthonormal bases with a controllable localization in the space–wave number domain, notably [19], [16], [18], [17]. In Numerical Analysis many ingredients of Calderón-Zygmund theory were used in the Fast Multipole algorithm for computing potential interactions [35], [24], [13]. The