Wavelets: A tool for time-frequency analysis
Summary form only given. In the simplest case, a family wavelets is generated by dilating and translating a single function of one variable: ha,b(x) = |a|-1/2h (x-b/a). The parameters a and b may vary continuously, or be restricted to a discrete lattice of values a = a0m, b = na0mb0. If the dilation and translation steps a0 and b0 are not too large, then any L2-function can be completely characterized by its inner products with the elements of such a discrete lattice of wavelets. Moreover, one can construct numerically stable algorithms for the reconstruction of a function from these inner products (the wavelet coefficients). For special choices of the wavelet h decomposition and reconstruction can be done very fast, via a tree algorithm. The wavelet coefficients of a function give a time-frequency decomposition of the function, with higher time-resolution for high-frequency than for low-frequency components. The analysis can easily be extended to higher dimensions. An especially important case is orthonormal bases of wavelets. It turns out that there exist functions h, with very good regularity and decay properties, such that the discrete lattice with a0 = 2 and b = 1 leads to an orthonormal set of functions hmn that spans all of L2(R). Such orthonormal bases are always associated with efficient tree algorithms.
