Here's a radical change of pace: this is a story. It comes doubly from Ingrid Daubechies. First she told the original version at a post-lecture dinner. It was so wonderful that I rushed for my keyboard at a rate that probably should have earned me a traffic ticket. But all my rushing could not prevent a certain amount of information from leaking or getting jostled, so my first draft was full of holes and places I was suspicious about. Ingrid kindly consented to take it and fix it up (Prof. A became Prof. Grossman, Paris became Philadelphia, sundry technical explanations got rescued from the scrambling maachine, etc.) Today I got her corrections, zapped them onto my computer and here's the tale:
"Where did the term wavelets come from?" seems an innocuous
question, though certainly a very reasonable one to bring up if one
happens to be at a dinner table with someone who has been packing the hall
for a series of lectures on her work in wavelet theory. Ingrid
Daubechies' reply, which began with the word itself and progressed through
the development of the whole theory, had every one of us on the edge of
our seat for fifteen minutes, and was far too fascinating to be left
untold. I shall therefore attempt to tell it:
The name really came from Geophysics, where for many years people had been using Fourier transforms, but in order to make them useful under the constraints of the field, had been setting up different sets of window functions frames, known as "Smith's wavelets" or "Jones's wavelets". Along came geophysicist Jean Morlet, of the oil company Elf Acquitaine, who felt that more could be accomplished with what he called "wavelets of constant shape." It cut no ice with his colleagues, but his conviction was such that he carried out a batch of computations, printed them out on miles of computer paper (remember when computer stuff came out in LONG zigzags?) and trundled them off to Prof. Alex Grossman, a physicist at the University of Marseilles. It was, according to Ingrid, an absolute mass of numbers, but Grossman somehow managed to see in them the significance of the idea and of various patterns that arose. He was intrigued, and his interest resulted in a formal framework for Morlet's computations.
The next chapter takes place in a line in Paris. It seems that at that time at the Ecole Polytechnique, the mathematicians and the physicists shared a photocopier--a slow one, at that. As a result, people spent a fair amount of time waiting for it. One day, an old friend of Alex Grossman found himself in front of his mathematical friend Prof. Yves Meyer, and struck up a conversation about these wavelets of constant shape. Meyer's first reaction was "Pooh, we've done similar transforms in harmonic analysis forever", but Grossman's and Morlet's papers succeeded in convincing him that looking from this new perspective did indeed produce new insights. They began collaborating, and having left the vicinity of Smith's wavelets and Jones's wavelets, felt no need to modify the word. It became the theory of wavelets (to the annoyance, one gathers, of a certain number of geophysicists).
At about this stage Ingrid, who did all of her degree work in abstract physics but came into mathematics in the process of making sense of it all, began working with wavelets. One of her papers was written in collaboration with Grossman, who in turn discussed some of its content with Yves Meyer . It became thereby the only paper Ingrid has written jointly with somebody she had never met. After it was written up, Meyer decided, as Ingrid put it, to read this paper he had just written. Coming at it from the outside of the field, he made an observation: all of the computations were done with a highly redundant set of wavelets, and it was clear that the implicit assumption within the field was that this was a necessity, i.e., that there could not exist an orthonormal basis. That sounded like an interesting idea, so he set about to prove that it was correct. There was a slight glitch, though, which caused his proof to break down. The glitch was that he found an orthonormal basis. A very neat one, in fact, because as he worked his way through the computations whole great chunks miraculously went to zero and quietly washed away.
That was a major break-through. So much so, in fact, that it was a natural topic of conversation when one of Meyer's students went for a walk on the beach with Stephane Mallat, an old friend from college who had gone on to work on a Ph.D. in computer vision. Mallat was fascinated and very excited, because he was sure that this mathematical construction was related to general multiscale considerations used in vision theory. Back from his vacation, he set about exploring his hunch seriously, and within a few months had enough of a theory put together to be bound and determined to discuss it with Meyer. Discovering that Meyer was giving a lecture series in Chicago, he found a cheap Philadelphia-Chicago flight and tracked him down there. Apparently he was pretty convincing, because Meyer abandoned his colleagues and the two of them holed up for three days in Zygmund's office, emerging with a whole new vision of wavelet theory. The implications of that vision were way beyond me, but Ingrid's description was that it resulted in the theory being far more organic, with a natural growth pattern among the results. A few weeks later, Meyer was back at a conference in the French Pyrenees, explaining the construction excitedly to colleagues, with back-of-the-envelope sketches. This, in turn, led to the construction of Ingrid's wavelet bases.
An upshot of Mallat's and Meyer's multiresolution vision which I absolutely loved was that in the derivation of Meyer's original orthonormal wavelet basis a whole bunch of situations where the best description of why a computation worked out as beautifully as it did was "And then a miracle occurs..." suddenly became reasonable and connected. I suspect that in the next generation of text books the operative phrase at those spots will be "It is clear that..."