Recent developments in the theory of Walsh series

The Decay of the Walsh Coefficients of Smooth Functions

“…, ω b−1 b }, where k is a nonnegative integer and ω b = e 2πi/b , were first introduced in [11] and further early results were obtained in [2,6]. See for example [9] for an overview. It is well known that Walsh functions form a complete orthonormal system of L 2 ([0, 1)), see [2,7].…”

On the divergence of Walsh-Fejér means of bounded functions on sets of measure zero

“…Kheladze [9] proved that for any 1 p < ∞ and every set of measure zero there exists a function in L p (0, 1) whose WalshFourier series diverges on that set. In his survey paper Wade [17] mentioned that it was still unknown whether for every set of measure zero there exists a bounded and continuous function whose Walsh Fourier series diverges on the set. Harris [7] proved that for every closed subset of the Vilenkin group of measure zero there exists a continuous (on the group) function whose Fourier series with respect to the Vilenkin system of bounded type diverges on the set.…”

Compact Sets of Divergence for Continuous Functions on a Vilenkin Group

“…It was expected that the same result would hold for WalshFourier series, but repeated attempts have failed to settle this question. (For a survey of these attempts see [4].) Thus far only particular uncountable sets of divergence have been identified.…”