Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-16T16:43:11.120Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[0]Assani, I.. A Wiener—Wintner property for the helical transform. Submitted.Google Scholar
[1]Assani, I.. The helical transform and the a.e. convergence of Fourier series.Google Scholar
[2]Assani, I. & Petersen, K.. The helical transform as a connection between ergodic theory and harmonic analysis. Trans. Amer. Math. Soc.Google Scholar
[3]Calderón, A.. Ergodic theory and translation-invariant operators. Proc. Nat. Acad. Sci. USA 59 (1968), 349–353.CrossRefGoogle ScholarPubMed
[4]Campbell, J. & Petersen, K.. The spectral measure and Hilbert transform of a measure preserving transformation. Trans. Amer. Math. Soc. 313 (1989), 121–130.CrossRefGoogle Scholar
[5]Carleson, L.. On convergence and growth of partial sums of Fourier series. Acta Math. 116 (1966), 135–157.CrossRefGoogle Scholar
[5']Castaing, C. & Valadier, M.. Convex Analysis and Measurable Multifunctions. Springer Lecture Notes in Mathematics 580. Springer Verlag, Berlin, 1977.CrossRefGoogle Scholar
[6]Cotlar, M.. A unified theory of Hilbert transforms and ergodic theorems. Rev. Mat. Cuyana 1 (1955), 105–167.Google Scholar
[6']Halmos, P.. Lectures on Ergodic Theory. Chelsea, New York, 1956.Google Scholar
[7]Hunt, R.. On the Convergence of Fourier Series, Orthogonal Expansions and Their Continuous Analogues. ed, Haimo, D. T.. S. Ill. Univ. Press, Carbondale, 1968, pp. 235–255.Google Scholar
[8]Ito, K. & Nisio, M.. On the convergence of sums of independent Banach space valued random variables. Osaka Math. J. 5 (1968), 35–48.Google Scholar
[9]Jajte, R.. On the existence of the ergodic Hilbert transform. Ann. Prob. 15 (1987), 831–835.CrossRefGoogle Scholar
[10]Kahane, J. P.. Some Random Series of Functions. 2nd ed.Cambridge University Press, Cambridge, 1985.Google Scholar
[11]Marcus, M. B., ξ-radial processes and random Fourier series. Memoirs Amer. Math. Soc. 181. Providence, RI, 1987.Google Scholar
[12]Stein, E. M.. On limits of sequences of operators. Ann. Math. 74 (1961), 140–170.CrossRefGoogle Scholar
[13]Veech, W. A.. Commentary on [41a, b]. Wiener, N.. Collected Works, Vol. 1 of the series, Mathematicians of Ourm Tie. MIT Press, Cambridge, MA, 1976.Google Scholar
[14]Wiener, N. & Wintner, A.. Harmonic analysis and ergodic theory. Amer. J. Math. 63 (1941), 415–426.CrossRefGoogle Scholar