The Wiener—Wintner property for the helical transform of the shift on [0, 1]ℤ

Abstract

AbstractLet ([0, 1]z, ℬ([0, 1]Z), μz, ϕ) be the dynamical system where ϕ is the shift on the product of the unit interval with Lebesgue measure. We show that this dynamical system has the following properties: (1) there exists ƒ ∈ L1(μz) (in fact in L(Log Log L)β) where 0 <β< 1 for which the following Wiener-Wintner property does not hold: (W-W). There exists a single null set N ⊂ X off which for all x ∈ X/N the sequence converges for all ε ⊂ [0,1).(2) The property (W-W) holds in all Lp(μz), 1< p≤∞. Added to a continuity property of the helical transform, (W-W) is equivalent to the Carleson-Hunt result on the pointwise convergence of Fourier series.