Uniqueness and summability of two-dimensional Walsh series and their generalizations

This paper explores a variety of questions associated with two-dimensional Vilenkin series and their special case two-dimensional Walsh series. Some of the results are an extension of known results of the one-dimensional case to those of two-dimensions. These theorems concern the uniqueness of two-dimensional Walsh series, and a Tauberian theorem,which shows the relationship between the summability and convergence of square partial sums of two-dimensional Walsh series. The final results generalize known theorems concerning the summability of two-dimensional Vilenkin-Fourier series of unbounded type. In all cases, the sequence p = (p_k) is the generating sequence for the Vilenkin group G_p (p_k = 2 in the Walsh case),and the sequence (P _k) is given by P₀ = 1 and P_k = p₀p₁ … p_k - 1.

Fundamental real analysis and measure theory are vital to the techniques used. First, a two-dimensional quasi-measure is defined and then each two-dimensional Vilenkin series is shown to have an associated two-dimensional quasi-measure. Using this last fact and newly defined two-dimensional derivates, a two dimensional variant is found to the classical result that differentiable functions with negative derivatives are decreasing. This leads to a uniqueness result which says that if a two-dimensional Walsh series, S, satisfy both a two-dimensional CS-condition for all χ ∈ G² and the condition

lim n %rarr; ∞ S_2ⁿ,2ⁿ (χ) = 0

for all but countably many χ ∈ G², then S is the zero series.

Next the dyadic square partial sums are found to be very good and finite or very bad and infinite, which leads to a Tauberian style theorem. It says that on a measurable subset of [0,1)² a two-dimensional Walsh series, S, with bounds on certain Cesaro means for each χ in the subset, then there is a function ƒ such that

lim n %rarr; ∞ S_2ⁿ,2ⁿ (χ) = ƒ(χ)

for almost every x in the measurable set.

The final set of results deals with growth estimates for two-dimensional Vilenkin-Fourier Series of unbounded type. Under The Condition of the sequence (p_k) known as δ-strongly quasi bounded,a certain maximal summability operator is bounded.