Multiple Trigonometric Series and Multiple Walsh Series

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.