Main areas are harmonic analysis and several complex variables.
Most of my work has arisen from work on a specific problem called the Pompeiu problem. This question asks about whether a function may be uniquely characterized by a certain set of integrals. If we consider integrals over a given set, S, as well as integrals over all translations and rotations of S, does this integral information uniquely characterize the function f?
This question can be interpreted as the issue of injectivity of a certain transform. Note that when moments are included in the integrals, differentials are introduced which allow us to characterize analytic functions, as well as other more general conclusions involving differentials.
My work with Dr. Berenstein (UMCP) and Dr. Chang (Georgetown) works with this problem in the setting of the Heisenberg group H^n. In this case, rigid motions are replaced by the action of translation by the Heisenberg group.
We investigated the moment version of this problem in the papers:
One limitation of this line of work is that so far only sets which are radial or polyradial have been considered. In these cases, rotations were not included. So these generalized the theorems of two radii to the Heisenberg group, but left much of the Pompeiu problem itself unaddressed. One future direction is to find a method to address the more general problem, for non-radial sets, and including rotations. We have begun on this work in the paper:
which begins to address the issue of rotations.
A logical extension of this direction of work includes the possibility to address the connection of the Pompeiu problem to a certain free boundary problem. We would like to see how to express such a connection in the Heisenberg setting.
Other work under way includes work on the local version of the Pomepiu problem for the Heisenberg group.
Once injectivity of the Pompeiu transform is established, one issue that remains to be considered is that of inversion of the transform. This question would correspond to recovering the function from the collection of integrals, and has been called deconvolution in the Euclidean setting. We will continue to investigate this issue in the Heisenberg setting.
Note that this issue relates to multi-channel signal detection, and to Hormander's strongly coprime condition. We worked on this question some in the setting where moments are included:
Much of the method in handling this problem come from fourier analysis. In the Heisenberg group, this was accomplished by use of exponential Laugerre functions, as well as a Gelfand transform. We also establish a connection of this work to the group Fourier transform for the Heisenberg group in:
This approach to the problem may extend to help us when
considering other aspects of the Pompeiu problem in
the Heisenberg setting.
Other avenues of research in this direction include the following:
Comparison of the Pompeiu problem in Clifford algebra setting and Heisenberg setting, and possible extension of the results,
Connection of Pompeiu problem to heat and wave equations,
Work on analogues of Hormander's strongly coprime condition for the Heisenberg setting.
Another important direction for the research includes looking at how to work on these problems for a more general setting than the Heisenberg group.
Some other new directions include the following:
Radon transform for the Heisenberg group
Partial differential equations
Hardy spaces and BMO
Also interested in other connections of my work to the field of several complex variables and to issues of analytic extension.
Also interested in relation to work on analytic disks.