ABSTRACT
Wavelets appear highly unlikely to have the revolutionary impact upon pure mathematics that Fourier analysis has had. "With wavelets it is possible to write much simpler proofs of some theorems," Daubechies says. "But I know of only a couple of theorems that have been proved with wavelets, that had not been proved before." (Strichartz adds that this simplification is in itself not negligible.) But wavelets are suited to a wide range of applications. It can be instructive to compare wavelet coefficients at different resolutions. Zero coefficients, which indicate no change, can be ignored, but nonzero coefficients indicate that something is going on, whether an error, an abrupt change in the signal, or noise (an unwanted signal that obscures the real message). If coefficients appear only at fine scales, they generally indicate the slight but rapid variations characteristic of noise. "The very fine-scale wavelets will try to follow the noise," Daubechies explains, while wavelets at coarser resolutions are too approximate to pick up such slight variations.
