Multiple Trigonometric Series and Multiple Walsh Series

Depaul University, Chicago IL

Investigators

Abstract

Proposal Abstract A square partial sum of a double trigonometric series is the sum of all the terms with both indices less than or equal to a fixed value. Our first goal is to study square uniqueness for double trigonometric series. By this we mean that if the sequence of square partial sums of a double trigonometric series converges to zero everywhere, then the series is necessarily the trivial series. If this is true, we will then try to generalize this result to higher dimensions. The corresponding statements for circular/spherical convergence have been shown by Shapiro in dimension 2 and by Bourgain in higher dimensions; and by Ash-Freiling-Rinne and, independently, Tetunashvili for the unrestrictedly rectangular convergence case in any dimension. There is some evidence that square uniqueness may actually be false. For example, there is an everywhere square convergent double trigonometric series with coefficients having faster than polynomial growth rate. Our second goal is to study a related question: uniqueness for multiple Walsh series under different types of summation modes. The new approach we will take is to use classical harmonic analysis methods. By combining the traditional martingale approach with the new techniques developed from recent progress made in the area of multiple trigonometric series, we expect much progress can be made here. This in turn may give insights into the square uniqueness question for trigonometric series. The third goal is to study the long standing open question about the pointwise circular convergence for Fourier series of square integrable functions. We will try to shed some light on this by studying the corresponding question for double Walsh series, which are special form of tree-index and two parameter martingales. 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 combined to make the general surface. Gathering the simple surfaces in different orders may lead to different resultant surfaces. For certain gathering procedures, uniqueness has been proved. That is, for such gathering, there is only one way to produce the resultant surface. The most important gathering procedure for which the uniqueness remains an open question is called square convergence. We will try to determine if uniqueness holds for this procedure. Another way to construct a surface is to build it up as a combination of two dimensional oscillating square waves. Such a process is called a multiple Walsh series. We will consider the uniqueness question in this context also. We hope that understanding one of the two methods of construction may lead to insights about the other.