Mathematical Sciences: Uniqueness of Multiple Trigonometric Series

Ash 9307242 This mathematical research focuses on problems of multiple trigonometric series. The single-variable theory of such series could be said to be reasonably well understood, although there remain a number of exceedingly difficult problems still unresolved. Multiple trigonometric series have intrinsic problems which do not allow for simple generalizations from the one-variable case. One of the primary issues of convergence results from the variety of natural ways one can try to sum a multiple series. Even the most basic question of whether a multiple series which sums to the zero function must have zero coefficients (Cantor's theorem in one variable). If unrestricted rectangular convergence is allowed then the answer is affirmative in all dimensions. This was only proved two years ago. For spherical and square convergence, the answer is still unknown, except for spherical in dimension two. Work will be done to extend the recent results obtained for rectangular convergence to the remaining cases. Related work will be carried out on the almost everywhere convergence of multiple Fourier series. For functions in the Lebesgue spaces with power less than two, spherical partial sums may diverge on sets of positive measure. Nothing is known about convergence for functions with finite quadratic norm (Hilbert space). To study this class, work will first concentrate on radial functions to see if counterexamples already exist within this group. The influence of trigonometric series on the development of modern mathematical analysis is impossible to record in this short space. The basic U.S. graduate courses are replete with byproducts of the research efforts that have gone into our attempts to understand these building blocks of mathematical synthesis used universally by all scientists. Yet the underlying summability properties of the series remains, to a large extent, an enigma. This project seeks to build on a remarkable breakthrough which occurred when four researchers settled one of the most famous unsolved problems of multiple Fourier series two years ago. ***