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C. (1992). Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. , 45, 485–560. 13. , and Vial, P. (1992). Multiresolution analysis, wavelets, and fast algorithms on an interval. C. R. Acad. Sci. Paris A, 316, 417–421. 14. Daubechies, I. (1992). Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, SIAM, Philadelphia. WAVELETS 15. Daubechies, I. (1994). Two recent results on wavelets: wavelet bases for the interval, and biorthogonal wavelets diagonalizing the derivative operator.
Much of wavelets’ potential in applications stems from the conjunction of three key features: 1. Wavelets provide sparse representations of a broad class of functions and signals. 2. Wavelets can achieve very good localization in both time and frequency. 3. There are fast algorithms for computing wavelet representations in practice. Here, sparsity refers to the fact that most of the coefficients in the linear combination are nearly zero. A single wavelet basis can provide sparse representations of many spaces of functions at the same time.
Given a coarse approximation P0 f , we can refine to the finer approximation PJ f for any J > 0 by adding the details from the successive Wj spaces. Specifically, J PJ f = P0 f + (Pj f − Pj−1 f ) j=1 J−1 = P0 f + Qj f , j=0 and as J → ∞, ∞ f = P0 f + Qj f . (13) j=0 This successive refinement exactly mimics the inhomogeneous wavelet representation: the coarse approximation P0 f corresponds to a linear combination of the φ0,k , and each Qj f for j 0 corresponds to a linear combination of the ψj,k .