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

Published online by Cambridge University Press:  19 September 2008

I. Assani
I. Assani
Affiliation:
Department of Mathematics, CB# 3250, Phillips Hall, University of North Carolina, Chapel Hill, NC 27599, USA
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Abstract

Let ([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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 4 , December 1992 , pp. 659 - 672
Copyright
Copyright © Cambridge University Press 1992

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