# American Institue of Mathematical Sciences

2009, 2(2): 325-336. doi: 10.3934/dcdss.2009.2.325

## The modulated ergodic Hilbert transform

 1 Department of Mathematics, North Dakota State University, P.O. Box 6050, Fargo, ND 58108-6050, United States

Received  February 2008 Revised  November 2008 Published  April 2009

In this article we study the ergodic Hilbert transform modulated by bounded sequences. We prove that sequences satisfying some variation conditions and are universally good for ordinary ergodic averages, such as the sequences defined by the Fourier coefficients of $L_p$ functions, are universally good modulating sequences for the ergodic Hilbert transform. We also prove that sequences belonging to the subfamily $B_1^{\alpha}$ of the two-sided bounded Besicovitch class $B_1$ are good modulating sequences for the ergodic Hilbert transform.
Citation: Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325
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