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: h_a,b(x)=|a|^-1/2 h (x-b/a). The parameters a and b may vary continuously, or be restricted to a discrete lattice of values a=a₀^m, b=na ₀^mB₀. 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