ON THE SUMMABILITY OF THE DISCRETE HILBERT TRANSFORM

Rashid A. Aliev     (Baku State University, Baku, AZ 1148; Institute of Mathematics and Mechanics, NAS of Azerbaijan, Baku, AZ 1141, Azerbaijan)
Aynur F. Amrahova     (Baku State University, Baku, AZ 1148, Azerbaijan)

Abstract


In this paper, we study the asymptotic behavior of the distribution function of the discrete Hilbert transform of  sequences from the class \(l_{1} \) and find a necessary condition and a sufficient condition for the summability of the discrete Hilbert transform of a sequence from the class \(l_{1} \).

Keywords


Discrete Hilbert transform, Asymptotic behavior of the distribution function, Class of summable sequences

Full Text:

PDF

References


  1. Andersen K.F. Inequalities with weights for discrete Hilbert transforms. Canad. Math. Bul., 1977. Vol. 20. P. 9–16.
  2. Belov Y., Mengestie T.Y., Seip K. Discrete Hilbert transforms on sparse sequences. Proc. London Math. Soc., 2011. Vol. 103, No. 1. P. 73–105. DOI: 10.1112/plms/pdq053
  3. Belov Y., Mengestie T.Y., Seip K. Unitary discrete Hilbert transforms. J. Anal. Math., 2010. Vol. 112. P. 383–393. DOI: 10.1007/s11854-010-0035-y
  4. De Carli L., Samad S. One-parameter groups and discrete Hilbert transform. Canad. Math. Bull., 2016. Vol. 59. P. 497–507. arXiv: 1506.03362 [math.FA]. URL: https://arxiv.org/pdf/1506.03362.pdf
  5. Gabisonija I., Meskhi A. Two weighted inequalities for a discrete Hilbert transform. Proc. A. Razmadze Math. Inst., 1998. Vol. 116. P. 107–122. URL: http://rmi.tsu.ge/proceedings/volumes/ps/v116-4.ps.gz
  6. Hunt R., Muckenhoupt B., Wheeden R. Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc., 1973. Vol. 176, No. 2. P. 227–251. DOI: 10.2307/1996205
  7. Laeng E. Remarks on the Hilbert transform and some families of multiplier operators related to it. Collect. Math., 2007. Vol. 58, No. 1. P. 25–44. URL: https://www.raco.cat/index.php/CollectaneaMathematica/article/view/57795
  8. Liflyand E. Weighted Estimates for the Discrete Hilbert Transform. In: Methods of Fourier Analysis and Approximation Theory. Applied and Numerical Harmonic Analysis, ed. M. Ruzhansky, S. Tikhonov. Cham: Birkhäuser, 2016. P. 59–69. DOI: 10.1007/978-3-319-27466-9_5
  9. Rakotondratsimba Y. Two weight inequality for the discrete Hilbert transform. Soochow J. Math., 1999. Vol. 25, No 4. P. 353–373. URL: http://mathlab.math.scu.edu.tw/mp/pdf/S25N44.pdf
  10. Riesz M. Sur les fonctions conjuguees. Math. Z., 1928. Vol. 27. P. 218–244. URL: https://eudml.org/doc/167977
  11. Stepanov V.D., Tikhonov S.Yu. Two weight inequalities for the Hilbert transform of monotone functions. Dokl. Math., 2011. Vol. 83, No. 2. P. 241–242. DOI: 10.1134/S1064562411020359



DOI: http://dx.doi.org/10.15826/umj.2018.2.002

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.