Mathematical Sciences: RUI Uniqueness of Multiple Trigonometric Series and Symmetric Analysis

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This project continues collaborative research on mathematical questions concerning multiple trigonometric series. The starting point of the work rests with the 1870 result of G. Cantor who showed that at most one trigonometric series can converge to a given function. This is a different question from that of determining whether a given series converges. It falls into the category of what is known as a uniqueness question. A longstanding problem has been the uniqueness question for multiple trigonometric series. There are many more obstacles to this question arising from the multiple methods of summation available, which do not show up in the one variable case. The investigators have determined that the uniqueness is valid in the case of rectangular summation. Work will now be done in carrying over newly developed techniques to investigate other summation methods such as iterated or spherical convergence. In addition, further studies will be carried out on questions of multiple uniqueness and the problem of determining when a trigonometric series (in several variables) is a Fourier series. Trigonometric series form the backbone of the modern theory of harmonic analysis. They play a ubiquitous role in the study of analytic problems ranging from pure number theory to the solution of partial differential equations of physics. Recent results on the uniqueness question has opened up a new avenue of research possibilities which have essentially been dormant for decades.