At both the level of simple analogy, and at the level of deep underlying techniques, probability and harmonic analysis are intimately related. This has been understood since the beginning development of probability theory. In the 1920s and 30s many analysts noticed that certain trigonometric series — lacunary series — behave like sums of independent random variables. This resulted in central limit theorems and laws of the iterated logarithm for lacunary series that mirror similar theorems in probability theory. This work was later expanded and refined throughout the 1940s and 50s. Later, in the 1960s and 1970s, analysts explored the connections between harmonic analysis and martingale theory from probability.
Much of the work I have done continues in this tradition. Together with co-authors, this includes developing a theory of the probabilistic behavior of certain functionals, so-called
Littlewood-Paley functions, that is analogous to previously established theory for martingales. This work first involves finding the correct analogues in harmonic analysis of certain objects in probability theory. What is exciting is that the techniques used to develop this theory involve a careful and subtle mix of ideas from analysis and probability.
In other work, with other co-authors, I have investigated some fundamental questions regarding a certain type of non-linear diffusion equation, the two phase Stefan problem: ut=Δα(u)ut=Δα(u) where α(u)=u+1α(u)=u+1 for u<−1u<−1, α(u)=0α(u)=0 for −1≤u≤1−1≤u≤1, and α(u)=u−1α(u)=u−1 for u>1u>1. This models the flow of heat within a substance which can be in a liquid phase or a solid phase, and for which there is a latent heat to initiate phase change. These types of problems model phase change in say, an ice-water mixture; the “free boundary” is the moving boundary between the water and the ice. As with any differential equation the first question one asks is about the existence and uniqueness of solutions. Our work shows existence and uniqueness of solutions under certain conditions, and regularity of solutions. There is still work to be done to find the most general conditions under which there is existence and uniqueness. Many other issues remain open: the rectifiability of the free boundary and uniqueness of solutions in the sense of measures. A deeper understanding of these is important not only for this specific problem, but is important in general in the study of nonlinear parabolic partial differential equations and free boundary problems. There is a lot of work left to be done to investigate the problem ut=Δα(u)ut=Δα(u) for other functions α.α.
I also have an interest in investigating the effect of sequence transformations on the sequence of partial sums of a Fourier series, with the hope of accelerating their convergence. Many Fourier series arising in practice, in particular, those of functions with jump discontinuities, converge slowly, so it is natural to try to apply extrapolation or convergence acceleration methods to speed their convergence. In work with students I have investigated the effects of the Aitken δ2δ2 process and the Lubkin -W transform on the partial sums of Fourier series. These are techniques from numerical analysis that are known to accelerate the convergence of some types of series. However, in the case of Fourier series these do not work well; in fact, using ideas from number theory, we were able to prove that these methods are always doomed to failure. In another investigation, we sought to understand if the smoothness of a function would be a factor in the applicability of sequence transformation methods to the partial sums of its Fourier series. Here also, we were able to show that these methods must fail even for smooth functions, and again, most interestingly, the problem was reduced to ideas from number theory, although not the same number theoretic ideas as in previous investigations. Thus, there is apparently a link between number theory and the study of sequence acceleration methods applied to Fourier series, but this connection is completely unclear and not understood. There is much room for investigation of these ideas