{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T16:28:32Z","timestamp":1778689712987,"version":"3.51.4"},"reference-count":28,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,8]],"date-time":"2022-10-08T00:00:00Z","timestamp":1665187200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/sym14102096","type":"journal-article","created":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T03:32:56Z","timestamp":1665459176000},"page":"2096","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Fractional Hilbert Transform of Generalized Functions"],"prefix":"10.3390","volume":"14","author":[{"given":"Naheed","family":"Abdullah","sequence":"first","affiliation":[{"name":"Department of Mathematics, Government Girls PostGraduate College, Quetta 08734, Pakistan"},{"name":"Department of Mathematics, University of Balochistan, Quetta 87550, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saleem","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Balochistan, Quetta 87550, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"159","DOI":"10.4099\/math1924.9.159","article-title":"Convergence of Boehmians","volume":"9","author":"Mikusinski","year":"1983","journal-title":"Jpn. J. Math. New Ser."},{"key":"ref_2","first-page":"319","article-title":"The support of Mikusi\u0144ski operators","volume":"176","author":"Boehme","year":"1973","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_3","first-page":"405368","article-title":"Note on Boehmians for class of optical Fresnel wavelet transforms","volume":"2012","year":"2012","journal-title":"J. Funct. Spaces Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1186\/1687-1847-2013-77","article-title":"An estimate of Sumudu transforms for Boehmians","volume":"2013","year":"2013","journal-title":"Adv. Differ. Equ."},{"key":"ref_5","first-page":"36","article-title":"An extension of certain integral transform to a space of Boehmians","volume":"17","year":"2015","journal-title":"J. Assoc. Arab. Univ. Basic Appl. Sci."},{"key":"ref_6","first-page":"16","article-title":"Some general properties of a fractional Sumudu transform in the class of Boehmians","volume":"43","author":"Agarwal","year":"2016","journal-title":"Kuwait J. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1080\/10652460008819239","article-title":"Hilbert transform for Boehmians","volume":"9","author":"Karunakaran","year":"2000","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1080\/10652460212899","article-title":"Boehmians and their Hilbert transforms","volume":"13","author":"Karunakaran","year":"2002","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_9","first-page":"69","article-title":"Natural transform for distribution and Boehmian spaces","volume":"4","author":"Loonker","year":"2013","journal-title":"Math. Eng. Sci. Aerosp. (MESA)"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1080\/10652460701510642","article-title":"Stieltjes transform for Boehmians","volume":"18","author":"Roopkumar","year":"2007","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_11","first-page":"75","article-title":"Mellin transform for Boehmians","volume":"4","author":"Roopkumar","year":"2009","journal-title":"Bull. Inst. Math. New Ser."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"942","DOI":"10.1007\/s11253-020-01834-6","article-title":"Quaternionic Fractional Fourier Transform for Boehmians","volume":"72","author":"Roopkumar","year":"2020","journal-title":"Ukr. Math. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1080\/10652469808819206","article-title":"Fractional Fourier transform of generalized functions","volume":"7","author":"Zayed","year":"1998","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"9477","DOI":"10.1002\/mma.5304","article-title":"Quaternion Fourier integral operators for spaces of generalized quaternions","volume":"41","author":"Baleanu","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1186\/s13662-021-03391-z","article-title":"Estimates and properties of certain q-Mellin transform on generalized q-calculus theory","volume":"2021","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1186\/s13660-020-02471-0","article-title":"q-Analogues and properties of the Laplace-type integral operator in the quantum calculus theory","volume":"2020","year":"2020","journal-title":"J. Inequal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"5354","DOI":"10.1002\/mma.5379","article-title":"Some remarks on short-time Fourier integral operators and classes of rapidly decaying functions","volume":"42","year":"2019","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/S0165-1684(01)00186-4","article-title":"Fractional analytic signals","volume":"82","author":"Cusmariu","year":"2002","journal-title":"Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1364\/OL.21.000281","article-title":"Fractional hilbert transform","volume":"21","author":"Lohmann","year":"1996","journal-title":"Opt. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1109\/97.704973","article-title":"Hilbert transform associated with the fractional Fourier transform","volume":"5","author":"Zayed","year":"1998","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5027907","DOI":"10.1155\/2022\/5027907","article-title":"The Fractional Hilbert Transform on the Real Line","volume":"2022","author":"Abdullah","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"6911","DOI":"10.1364\/AO.37.006911","article-title":"Analysis of the fractional Hilbert transform","volume":"37","author":"Davis","year":"1998","journal-title":"Appl. Opt."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Deng, L., Hou, Z., Liu, H., and Sun, Z. (2019, January 27\u201330). A fractional hilbert transform order optimization algorithm based DE for bearing health monitoring. Proceedings of the 2019 Chinese Control Conference (CCC), Guangzhou, China.","DOI":"10.23919\/ChiCC.2019.8865403"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6620","DOI":"10.1364\/AO.36.006620","article-title":"Optical implementation of the fractional Hilbert transform for two-dimensional objects","volume":"36","author":"Lohmann","year":"1997","journal-title":"Appl. Opt."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Sharma, N. (2019, January 12\u201314). Design of fractional hilbert transform (\u03c0\/4). Proceedings of the 2019 3rd International Conference on Electronics, Communication and Aerospace Technology (ICECA), Coimbatore, India.","DOI":"10.1109\/ICECA.2019.8821968"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1186\/s13662-018-1644-5","article-title":"A fractional Fourier integral operator and its extension to classes of function spaces","volume":"2018","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_27","unstructured":"Rudin, W. (1987). Real and Complex Analysis, McGraw-Hill."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Zayed, A.I. (2019). Handbook of Function and Generalized Function Transformations, CRC Press.","DOI":"10.1201\/9780138752859"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2096\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:48:12Z","timestamp":1760143692000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2096"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,8]]},"references-count":28,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14102096"],"URL":"https:\/\/doi.org\/10.3390\/sym14102096","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202209.0029.v1","asserted-by":"object"}]},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,8]]}}}