Approximation using superposition of functions has existed since the early 1800's, when Joseph Fourier discovered that he could superpose sines and cosines to represent other functions. However, in wavelet analysis, the scale that one uses in looking at data plays a special role. Wavelet algorithms process data at different scales or resolutions. For many decades, scientists have wanted more appropriate functions than the the sines and cosines which comprise the bases of Fourier analysis, to approximate choppy signals. But with wavelet analysis, we can use approximating functions that are contained neatly in finite domains.
Wavelets are well-suited for approximating data with sharp discontinuities.
And if you further choose the best wavelets adapted to your data, or truncate the coefficients below a threshold, your data is sparsely represented. FTP a gzipped, postscript copy. W avelet Tonebursts Victor Wickerhauser has suggested that sound synthesis is a natural use of wavelets. If one wishes to approximate the sound of a musical instrument, the notes can be decomposed into its wavelet packet coefficients. Reproducing the note would then require reloading those coefficients into a wavelet packet generator and playing back the result.