Popular presentations of Carleson's results

Convergence of Fourier series

One way of describing Carleson's convergence theorem for the Fourier series of a quadratic integrable function is through sound waves. We can identify a function with a sound, in that the graph of the function describes the vibrations in a membrane. A complex function will normally make a rather noisy sound, whereas a smooth and regular wave function will yield a tone that is clear as a bell, like the sound of a tuning fork.

In this context, the sound from musical instruments may be represented as particular compositions of these smooth, pure tuning-fork waves, known as harmonic vibrations in mathematical terminology. The instruments' overtone patterns, or their individual sound pictures, are characteristic sums of pure harmonic vibrations with wavelengths that are integer products of a fundamental wavelength. Some wind instruments have relatively few prominent overtones, whereas the violin, for example, is specifically distinguished by its great abundance of overtones.

Fourier's problem can be formulated in this setting as follows: Can an orchestra, if possible with an infinite number of random small instruments, play every conceivable sound?

With a relatively high degree of approximation, we can say that Carleson in his work from 1966 gives an affirmative answer to this question and that he actually furnishes a rigorous mathematical proof for his assertion. The figure provides a little insight into how this process of approximation takes place. In this case, we start with a function, illustrated by the dotted line, that has a value of 1 in the interval from 0 to π and of –1 between π and 2π. The first approximation is by a pure harmonic vibration, designated with N=1. We see that this approximation is not very good. If we include 5 terms, drawn in as the curve N=5, we see that the approximation becomes much better. We can continue in this way; the more terms we include, the more the new curve will resemble the original step function curve.

The precise formula for the N=5 curve in this example is

and we can easily imagine how we can continue the approximation with more and more terms. An interesting point with this curve in particular is the two “horns” that develop just to the right of x=0 and just to the left of x=π. It turns out that no matter how many terms we use we will not be able to get rid of them. They will gradually become more and more slender, but they will not disappear. This phenomenon is called the Gibbs phenomenon and turns up in a number of similar examples.

We shall not attempt the proof of Carleson's convergence result here; we take the committee's warning seriously: "The proof of this result is so difficult that for over 30 years it stood mostly isolated from the rest of harmonic analysis. It is only within the past decade that mathematicians have understood the general theory of operators into which this theorem fits and have started to use Carleson’s powerful ideas in their own work."

HomeNews ArchiveCalendar      Editor: Anne Marie Astad  The Norwegian Academy of Science and Letters  E-mail: dnva@online.no