Compact Sets of Divergence for Continuous Functions on a Vilenkin Group

Abstract

ABSTRACT. Let G be a Vilenkin group. Let E C G be closed with Haar measure zero. We show there is a continuous function whose Vilenkin-Fourier series diverges at every point in E.

An Analogy of the Carleson–Hunt Theorem with Respect to Vilenkin Systems

“…we observe by (25) that this series converges in L p (G m ) norm. Hence f ∈ L p (G m ) and this series is the Vilenkin-Fourier series of f .…”

“…Later Schipp [44,49] proved that there exists an integrable function whose Walsh-Fourier series diverges everywhere. Kheladze [29,30] proved that for any set of measure zero there exists a function in f ∈ L p (G m ) (1 < p < ∞) whose Vilenkin-Fourier series diverges on the set, while the result for continuous or bounded function was proved by Harris [25] or Bitsadze [6]. Moreover, Simon [53] constructed an integrable function such that its Vilenkin-Fourier series diverges everywhere.…”

On Exceptional Sets of the Hilbert Transform

“…Concerning to Problem 2 we note that Harris [6] has proved that for any compact null set e ⊂ [0, 1] there exists a continuous function, whose Walsh-Fourier series diverges at any x ∈ e.…”

Book Review: Walsh series and transforms

“…But, given an ^ set E of measure zero there is a function / continuous on the dyadic group whose Walsh-Fourier series diverges everywhere on E (Harris [30]). It is an open question whether this result holds for all Borel sets E of measure zero.…”