Uniqueness for Multiple Trigonometric Series

ABSTRACT Ash-Wang Ash and Wang first major goal is to generalized theorems of Victor Shapiro and Jean Bourgain concerning uniqueness of representation by spherically convergent multiple trigonometric series. Shapiro proved that if a multiple trigonometric series is everywhere Abel spherically summable to an integrable function and if Shapiro's condition holds, then it is the Fourier series of that function. Shapiro's condition states that the ratio of the sum of the absolute values of the coefficients lying in an annulus of unit thickness to the radius tends to zero as the radius tends to infinity. A more natural condition is Connes' condition: that the sum of the squares of the coefficients lying on the surface of a sphere tends to zero as the radius tends to infinity. Since Connes' condition is a consequence of everywhere convergence, Bourgain was able to avoid any coefficient growth assumptions when he proved that a multiple trigonometric series everywhere spherically convergent to zero is the zero function. Ash and Wang will try to prove Shapiro's result with Connes' condition replacing Shapiro's condition in the hypothesis. A corollary of this theorem would be spherical uniqueness for trigonometric series that converge everywhere to an integrable function. Furthermore, Ash and Wang would like to lighten the hypothesis of everywhere convergence, by allowing an exceptional set on which convergence is not assumed. Such a set is called a set of uniqueness. Ash and Wang would like to show that all countable sets and certain uncountable sets are sets of uniqueness. Almost any surface is composed of simpler ones by a process called multiple Fourier analysis. A major long standing problem in pure mathematics is to show that this construction can be accomplished in only one way. This is called the problem of uniqueness. There are about a half dozen main varieties of this problem depending on just how the simpler surfaces are c ombined to make the general surface. In particular, Ash and Wang will try to determine if uniqueness holds for square convergent double trigonometric series. Ash and Wang will also try to show that certain thin sets may be ignored when considering the question of uniqueness for spherically convergent multiple trigonometric series. Since we live in a four dimensional world of space and time, it is also necessary to study a higher dimensional version of a surface. Such an object is called a manifold. Just as surfaces are associated with double trigonometric series, manifolds are associated with multiple trigonometric series. Thus, Ash and Wang will also try to determine if uniqueness holds for spherically convergent multiple trigonometric series.