Wavelets and Fast Numerical Algorithms

Numerical algorithms using wavelet bases are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in the new system of coordinates. As in all transform methods, such approach seeks an advantage in that the computation is faster in the new system of coordinates than in the original domain. However, due to the recursive definition of wavelets, their controllable localization in both space and wave number (time and frequency) domains, and the vanishing moments property, wavelet based algorithms exhibit a number of new and important properties. In the usual transform methods, the functions of the basis (e.g. exponentials, Chebyshev polynomials, etc.) are chosen to be eigenfunctions of some differential operator (e.g. solutions of the Sturm-Liouville problem). The choice of the differential operator and, hence, of the basis functions, is dictated by the availability of fast algorithms for expanding an arbitrary function into the basis. Unfortunately, classes of operators which have a sparse representation in such bases are very narrow. Wavelets, on the other hand, are not solutions of a differential equation. These functions are defined recursively and are generated via an iterative algorithm. They are translations and dilations of a single function . Instead of diagonalizing some differential operator, representations in the wavelet bases reduce a wide class of operators to a sparse form. Here the orthogonality of wavelets to the low degree polynomials (the vanishing moments property) plays a crucial role . The orthonormal bases of wavelets were fisrt constructed by Stromberg [33] and then by Meyer [25]. Later, the notion of the Multiresolution Analysis was introduced by Meyer [26] and Mallat [23]. Orthonormal bases of compactly supported wavelets were constructed by Daubechies [16]. There are many new constructions of orthonormal bases with a controllable localization in the time–frequency domain, notably ”wavelet-packet” bases in [13] and [15], local trigonometric bases in [14] and [24], wavelet bases on the interval in [11], [12] and [22]. Very important connection exists between the wavelets and the technique of subband coding in signal processing. In fact, the discrete wavelet