Over the past decade a number of mathematicians accustomed to the abstractions of pure research have been dirtying their hands—with great enthusiasm—on a surprising range of practical projects. What these tasks have in common is a new mathematical language, its alphabet consisting of identical squiggles called wavelets, appropriately stretched, squeezed, or moved about.
A whole range of information—your voice, your fingerprints, a snapshot, x-rays ordered by your doctor, radio signals from outer space, seismic waves—can be translated into this new language, which emerged independently in a number of different fields, and in fact was only recently understood to be a single language. In many cases this transformation into wavelets makes it easier to transmit, compress, and analyze information or to extract information from surrounding "noise"—even to do faster calculations.
In their initial excitement some researchers thought wavelets might virtually supplant the much older and very powerful mathematical language of Fourier analysis, which you use every time you talk on the telephone or turn on a television. But now they see the two as complementary and are exploring ways to combine them or even to create more languages "beyond wavelets."
Different languages have different strengths and weaknesses, points out Meyer, one of the founders of the field: "French is effective for analyzing things, for precision, but bad for poetry and conveying emotion—perhaps that's why the French like mathematics so much. I'm told by friends who speak Hebrew that it is much more expressive of poetic images. So if we have information, we need to think, is it best expressed in French? Hebrew? English? The Lapps have 15 different words for snow, so if you wanted to talk about snow, that would be a good choice."
Some information processing is best done in the language of Fourier; other with wavelets; and yet other tasks might require new languages. For the first time in a great many years—almost two centuries, if one goes back to the very birth of Fourier analysis—there is a choice.
Although wavelets represent a departure from Fourier analysis, they are also a natural extension of it: the two languages clearly belong to the