Abstract
Sufficient conditions have been given for the convergence in norm and a.e. of the ergodic Hilbert transform (Gaposhkin in Theory Probab Appl 41:247–264, 1996; Cohen and Lin in Characteristic functions, scattering functions and transfer functions, pp 77–98, Birkhäuser, Basel, 2009; Cuny in Ergod Theory Dyn Syst 29:1781–1788, 2009). Here we apply these conditions to the rotated ergodic Hilbert transform \({\sum_{n=1}^\infty \frac{\lambda^n}{n} T^nf}\) , where λ is a complex number of modulus 1. When T is a contraction in a Hilbert space, we show that the logarithmic Hausdorff dimension of the set of λ’s for which this series does not converge is at most 2 and give examples where this bound is attained.
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References
Assani I.: Wiener–Wintner dynamical systems. Ergod. Theory Dyn. Syst. 23, 1637–1654 (2003)
Assani I.: Spectral characterization of Wiener–Wintner dynamical systems. Ergod. Theory Dyn. Syst. 24, 347–365 (2004)
Assani I., Lin M.: On the one-sided ergodic Hilbert transform. Contemp. Math. 430, 21–39 (2007)
Campbell J.: Spectral analysis of the ergodic Hilbert transform. Indiana Univ. Math. J. 35, 379–390 (1986)
Carleson L.: Selected problems on exceptional sets. Van Nostrand Mathematical Studies, vol. 13 D. Van Nostrand, New York (1967)
Cohen, G., Lin, M.: The one-sided ergodic Hilbert transform of normal contractions. In: Characteristic Functions, Scattering Functions and Transfer Functions, pp. 77–98. Birkhäuser, Basel (2009)
Cuny C.: On the a.s. convergence of the one-sided ergodic Hilbert transform. Ergod. Theory Dyn. Syst. 29, 1781–1788 (2009)
Cuny, C.: Pointwise ergodic theorems with rate and application to limit theorem for stationary processes, preprint (see arXiv:0904.0185v1)
Cuzick J., Lai T.L.: On random Fourier series. Trans. Am. Math. Soc. 261, 53–80 (1980)
Doob J.L.: Stochastic Processes. Wiley, New York (1953)
Falconer, K.: Fractal geometry. Mathematical Foundations and Applications, xxii+288 pp. Wiley, Chichester (1990)
Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153, xiv+676 pp. Springer, New York (1969)
Gaposhkin V.: Spectral criteria for existence of generalized ergodic transforms. Theory Probab. Appl. 41, 247–264 (1996)
Halmos P.R.: A nonhomogeneous ergodic theorem. Trans. Am. Math. Soc. 66, 284–288 (1949)
del Junco A., Rosenblatt J.: Counterexamples in ergodic theory and number theory. Math. Ann. 245(3), 185–197 (1979)
Kahane J.P.: Some Random Series of Functions, 2nd edn. Cambridge University Press, Cambridge (1985)
Lacey M., Terwilleger E.: A Wiener–Wintner theorem for the Hilbert transform. Ark. Mat. 46(2), 315–336 (2008)
Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)
Riesz, F., Nagy, B.Sz.: Functional Analysis. Translated from the French by Leo F. Boron. Dover, New York (1990)
Wiener N., Wintner A.: Harmonic analysis and ergodic theory. Am. J. Math. 63, 415–426 (1941)
Zygmund, A.: Trigonometric series, vol. I, II, corrected 2nd edn. Cambridge University Press, Cambridge (1968)
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Chevallier, N., Cohen, G. & Conze, JP. On the convergence of the rotated one-sided ergodic Hilbert transform. Positivity 15, 253–270 (2011). https://doi.org/10.1007/s11117-010-0070-z
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DOI: https://doi.org/10.1007/s11117-010-0070-z