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The Fractional Hilbert Transform of Generalised Functions
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Version 1
: Received: 23 August 2022 / Approved: 2 September 2022 / Online: 2 September 2022 (02:54:50 CEST)
A peer-reviewed article of this Preprint also exists.
Abdullah, N.; Iqbal, S. The Fractional Hilbert Transform of Generalized Functions. Symmetry 2022, 14, 2096. Abdullah, N.; Iqbal, S. The Fractional Hilbert Transform of Generalized Functions. Symmetry 2022, 14, 2096.
Abstract
The fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its extensive use in optics, engineering, and signal processing. In the present work, we aimed to expand the fractional Hilbert transform to a space of generalized functions known as Boehmians. We introduce a new fractional convolution operator for the fractional Hilbert transform to prove a convolution theorem similar to the classical Hilbert transform and also to extend the fractional Hilbert transform to Boehmians. We also construct a suitable Boehmian space on which the fractional Hilbert transform exists. Further, we investigate convergence of the fractional Hilbert transform for the class of Boehmians and discuss the continuity of the extended fractional Hilbert transform.
Keywords
Convolution; Boehmian; fractional Hilbert transform; Hilbert transform; equivalence class; delta sequences; compact support
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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