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The Fractional Hilbert Transform on the Real Line

Author

Listed:
  • Naheed Abdullah
  • Saleem Iqbal
  • Asma Khalid
  • Amnah S. Al Johani
  • Ilyas Khan
  • Abdul Rehman
  • Mulugeta Andualem
  • Krzysztof Puszynski

Abstract

This paper examines some special properties and important results of the fractional Hilbert transform (FHT) on the real line ℠. In this rigorous study, we modify certain theorems of classical Hilbert transform for FHT and develop new theorems. Moreover, FHT of some common functions is given and eigenfunctions are also studied. We prove that FHT, denoted by Hα, is an isomorphism on L2℠. We also show that in L2℠, FHT is an isometry. Furthermore, we investigate Riesz inequality on Lp℠for p>1 to establish Hilbert formulae for FHT.

Suggested Citation

  • Naheed Abdullah & Saleem Iqbal & Asma Khalid & Amnah S. Al Johani & Ilyas Khan & Abdul Rehman & Mulugeta Andualem & Krzysztof Puszynski, 2022. "The Fractional Hilbert Transform on the Real Line," Mathematical Problems in Engineering, Hindawi, vol. 2022, pages 1-11, April.
    • Handle: RePEc:hin:jnlmpe:5027907
    DOI: 10.1155/2022/5027907
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