The Fractional Hilbert Transform of Generalized Functions
The fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its widespread application in optics, engineering, and signal processing. In the present work, we expand the fractional Hilbert transform that displays an odd symmet...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/10/2096 |
| _version_ | 1797469871768862720 |
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| author | Naheed Abdullah Saleem Iqbal |
| author_facet | Naheed Abdullah Saleem Iqbal |
| author_sort | Naheed Abdullah |
| collection | DOAJ |
| description | The fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its widespread application in optics, engineering, and signal processing. In the present work, we expand the fractional Hilbert transform that displays an odd symmetry to a space of generalized functions known as Boehmians. Moreover, we introduce a new fractional convolutional operator for the fractional Hilbert transform to prove a convolutional theorem similar to the classical Hilbert transform, and also to extend the fractional Hilbert transform to Boehmians. We also produce a suitable Boehmian space on which the fractional Hilbert transform exists. Further, we investigate the convergence of the fractional Hilbert transform for the class of Boehmians and discuss the continuity of the extended fractional Hilbert transform. |
| first_indexed | 2024-03-09T19:27:12Z |
| format | Article |
| id | doaj.art-caab3e2fed644984adfa930ca6db16dd |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T19:27:12Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-caab3e2fed644984adfa930ca6db16dd2023-11-24T02:52:19ZengMDPI AGSymmetry2073-89942022-10-011410209610.3390/sym14102096The Fractional Hilbert Transform of Generalized FunctionsNaheed Abdullah0Saleem Iqbal1Department of Mathematics, Government Girls PostGraduate College, Quetta 08734, PakistanDepartment of Mathematics, University of Balochistan, Quetta 87550, PakistanThe fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its widespread application in optics, engineering, and signal processing. In the present work, we expand the fractional Hilbert transform that displays an odd symmetry to a space of generalized functions known as Boehmians. Moreover, we introduce a new fractional convolutional operator for the fractional Hilbert transform to prove a convolutional theorem similar to the classical Hilbert transform, and also to extend the fractional Hilbert transform to Boehmians. We also produce a suitable Boehmian space on which the fractional Hilbert transform exists. Further, we investigate the convergence of the fractional Hilbert transform for the class of Boehmians and discuss the continuity of the extended fractional Hilbert transform.https://www.mdpi.com/2073-8994/14/10/2096convolutionBoehmianfractional Hilbert transformHilbert transformequivalence classdelta sequences |
| spellingShingle | Naheed Abdullah Saleem Iqbal The Fractional Hilbert Transform of Generalized Functions Symmetry convolution Boehmian fractional Hilbert transform Hilbert transform equivalence class delta sequences |
| title | The Fractional Hilbert Transform of Generalized Functions |
| title_full | The Fractional Hilbert Transform of Generalized Functions |
| title_fullStr | The Fractional Hilbert Transform of Generalized Functions |
| title_full_unstemmed | The Fractional Hilbert Transform of Generalized Functions |
| title_short | The Fractional Hilbert Transform of Generalized Functions |
| title_sort | fractional hilbert transform of generalized functions |
| topic | convolution Boehmian fractional Hilbert transform Hilbert transform equivalence class delta sequences |
| url | https://www.mdpi.com/2073-8994/14/10/2096 |
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