American Institute of Mathematical Sciences

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

The modulated ergodic Hilbert transform

Doǧan Çömez 1,

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.
Keywords: Hilbert transform, Fourier coefficients, Besicovitch sequences..
Mathematics Subject Classification: Primary: 28D05; Secondary: 42B20, 40A3.
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
[1]

Nam Yul Yu. A Fourier transform approach for improving the Levenshtein's lower bound on aperiodic correlation of binary sequences. Advances in Mathematics of Communications, 2014, 8 (2) : 209-222. doi: 10.3934/amc.2014.8.209
[2]

Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579
[3]

Juan H. Arredondo, Francisco J. Mendoza, Alfredo Reyes. On the norm continuity of the hk-fourier transform. Electronic Research Announcements, 2018, 25: 36-47. doi: 10.3934/era.2018.25.005
[4]

Huichi Huang. Fourier coefficients of $\times p$-invariant measures. Journal of Modern Dynamics, 2017, 11: 551-562. doi: 10.3934/jmd.2017021
[5]

Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023
[6]

Michael Music. The nonlinear Fourier transform for two-dimensional subcritical potentials. Inverse Problems & Imaging, 2014, 8 (4) : 1151-1167. doi: 10.3934/ipi.2014.8.1151
[7]

Jan-Cornelius Molnar. On two-sided estimates for the nonlinear Fourier transform of KdV. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3339-3356. doi: 10.3934/dcds.2016.36.3339
[8]

Matti Viikinkoski, Mikko Kaasalainen. Shape reconstruction from images: Pixel fields and Fourier transform. Inverse Problems & Imaging, 2014, 8 (3) : 885-900. doi: 10.3934/ipi.2014.8.885
[9]

Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure & Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007
[10]

Rakesh Pilkar, Erik M. Bollt, Charles Robinson. Empirical mode decomposition/Hilbert transform analysis of postural responses to small amplitude anterior-posterior sinusoidal translations of varying frequencies. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1085-1097. doi: 10.3934/mbe.2011.8.1085
[11]

Andrey Kochergin. A Besicovitch cylindrical transformation with Hölder function. Electronic Research Announcements, 2015, 22: 87-91. doi: 10.3934/era.2015.22.87
[12]

Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform. Kinetic & Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027
[13]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
[14]

Sean Holman, Plamen Stefanov. The weighted Doppler transform. Inverse Problems & Imaging, 2010, 4 (1) : 111-130. doi: 10.3934/ipi.2010.4.111
[15]

James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013
[16]

Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103
[17]

Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko. On a class of model Hilbert spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5067-5088. doi: 10.3934/dcds.2013.33.5067
[18]

Ilesanmi Adeboye, Harrison Bray, David Constantine. Entropy rigidity and Hilbert volume. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1731-1744. doi: 10.3934/dcds.2019075
[19]

E. Camouzis, H. Kollias, I. Leventides. Stable manifold market sequences. Journal of Dynamics & Games, 2018, 5 (2) : 165-185. doi: 10.3934/jdg.2018010
[20]

Frank Fiedler. Small Golay sequences. Advances in Mathematics of Communications, 2013, 7 (4) : 379-407. doi: 10.3934/amc.2013.7.379

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences