Sets of uniqueness in compact, 0-dimensional metric groups

Abstract

ABSTRACT. An investigation is made of sets of uniqueness in a compact 0-dimensional space. Such sets are defined by pointwise convergence of sequences of functions that generalize partial sums of trigonometric series on Vilenkin groups. Several analogs of classical uniqueness theorems are proved, including a version of N. Bary's theorem on countable unions of closed sets of uniqueness.

Uniqueness of Almost Everywhere Convergent Vilenkin Series

“…Theorem 1 is known when f is a finite-valued, integrable function and (2) holds off a countable set rather than almost everywhere (see D. J. Grubb [3], and Bokaev and Skvortsov [1]). Grubb [3] has also shown that if S P n → 0 almost everywhere on G and (3) holds everywhere on G, then S is the zero series. Clearly, Theorem 1 contains all these results when f ∈ L q , q > 1, i.e., shows that uniqueness holds under mild growth conditions for simultaneously almost everywhere convergence and nonzero limits.…”