{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:20:17Z","timestamp":1760149217932,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,15]],"date-time":"2023-07-15T00:00:00Z","timestamp":1689379200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at King Khalid University","award":["RGP2\/237\/44"],"award-info":[{"award-number":["RGP2\/237\/44"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this study, the Wigner\u2013Ville distribution is associated with the one sided Clifford\u2013Fourier transform over Rn, n = 3(mod 4). Accordingly, several fundamental properties of the WVD-CFT have been established, including non-linearity, the shift property, dilation, the vector differential, the vector derivative, and the powers of \u03c4\u2208Rn. Moreover, powerful results on the WVD-CFT have been derived such as Parseval\u2019s theorem, convolution theorem, Moyal\u2019s formula, and reconstruction formula. Eventually, we deduce a directional uncertainty principle associated with WVD-CFT. These types of results, as well as methodologies for solving them, have applications in a wide range of fields where symmetry is crucial.<\/jats:p>","DOI":"10.3390\/sym15071421","type":"journal-article","created":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T00:35:04Z","timestamp":1689554104000},"page":"1421","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Wigner\u2013Ville Distribution Associated with Clifford Geometric Algebra Cln,0, n=3(mod 4) Based on Clifford\u2013Fourier Transform"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3369-0883","authenticated-orcid":false,"given":"Mohammad Younus","family":"Bhat","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Islamic University of Science and Technology, Kashmir 192122, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shahbaz","family":"Rafiq","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Islamic University of Science and Technology, Kashmir 192122, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3305-7340","authenticated-orcid":false,"given":"Mohra","family":"Zayed","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"81","DOI":"10.12988\/ijma.2014.311290","article-title":"Product theorem for quaternion Fourier transform","volume":"8","author":"Bahri","year":"2014","journal-title":"Int. J. Math. Anal."},{"key":"ref_2","first-page":"1","article-title":"Miyachi\u2019s theorem for the quaternion Fourier transform","volume":"39","author":"Haoui","year":"2019","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2511","DOI":"10.1016\/j.sigpro.2008.04.012","article-title":"Fractional quaternion Fourier transform, convolution and correlation","volume":"88","author":"Xu","year":"2008","journal-title":"Signal Process."},{"key":"ref_4","unstructured":"Brackx, F., Delanghe, R., and Sommen, F. (1982). Clifford Analysis, Pitman Advanced Pub. Program."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1007\/s00006-008-0098-3","article-title":"Clifford\u2013Fourier transform on multivector fields and Uncertainty principles for dimensions n = 2(mod 4) and n = 3(mod 4)","volume":"18","author":"Hitzer","year":"2018","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1109\/TVCG.2005.54","article-title":"Clifford\u2013Fourier transform on Vector Fields","volume":"11","author":"Ebling","year":"2005","journal-title":"IEEE Trans. Visual Comp. Graph."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s00006-006-0003-x","article-title":"Clifford\u2013Fourier transformations and uncertainty principle for the Clifford geometric algebra Cl3,0","volume":"16","author":"Bahri","year":"2006","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_8","unstructured":"Hitzer, E., and Bahri, M. (2006). Uncertainty Principle for the Clifford Geometric Algebra Cln,0, n = 3(mod 4) Based on Clifford\u2013Fourier Transform, Springer (SCI) Book Series Applied and Numerical Harmonic Analysis."},{"key":"ref_9","first-page":"125","article-title":"Convolution theorems for Clifford\u2013Fourier transform and properties","volume":"20","author":"Bahri","year":"2014","journal-title":"J. Indian Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/s00041-005-4079-9","article-title":"The Clifford\u2013Fourier transform","volume":"11","author":"Brackx","year":"2005","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2215","DOI":"10.1007\/s00006-016-0687-5","article-title":"General Steerable two-sided Clifford\u2013Fourier transform, Convolution and Mustard Convolution","volume":"27","author":"Hitzer","year":"2017","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1007\/s00006-015-0617-y","article-title":"A modified uncertainty principle for two sided quaternion Fourier transform","volume":"26","author":"Bahri","year":"2016","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_13","first-page":"3795120","article-title":"A variation on uncertainty principle and logarithmic uncertainty principle for continuous quaternion wavelet transform","volume":"217","author":"Bahri","year":"2017","journal-title":"Abstr. Appl. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2882","DOI":"10.1109\/78.542448","article-title":"On generalized marginal time-frequency distributions","volume":"44","author":"Xia","year":"1996","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3111","DOI":"10.1109\/TSP.2005.851134","article-title":"Wigner distributions and ambiguity functions of 2-D quaternionic and monogenic signals","volume":"53","author":"Hahn","year":"2005","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"139471","DOI":"10.1155\/2014\/139471","article-title":"On two dimensional quaternion Wigne-Ville distribution","volume":"2014","author":"Bahri","year":"2014","journal-title":"J. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Bhat, M.Y., Dar, A.H., Nurhidayat, I., and Pinelas, S. (2023). An Interplay of Wigner\u2013Ville Distribution and 2D Hyper-Complex Quadratic-Phase Fourier Transform. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7020159"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2779","DOI":"10.1007\/s11760-023-02495-1","article-title":"Quadratic-phase scaled Wigner distribution: Convolution and correlation","volume":"17","author":"Bhat","year":"2023","journal-title":"Signal Image Video Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"170213","DOI":"10.1016\/j.ijleo.2022.170213","article-title":"Wigner Distribution and Associated Uncertainty Principles in the Framework of Octonion Linear Canonical Transform","volume":"272","author":"Dar","year":"2022","journal-title":"Optik"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"786","DOI":"10.1515\/dema-2022-0175","article-title":"Wigner\u2013Ville Distribution and Ambiguity Function Associated with the Quaternion Offset Linear Canonical Transform","volume":"55","author":"Bhat","year":"2022","journal-title":"Demonstr. Math."},{"key":"ref_21","first-page":"217","article-title":"The Wigner distribution\u2014A tool for time-frequency signal analysis\u2014Part I: Continuous-time signals","volume":"35","author":"Claasen","year":"1980","journal-title":"Philips J. Res."},{"key":"ref_22","first-page":"276","article-title":"The Wigner distribution\u2014A tool for time-frequency signal analysis\u2014Part II: Discrete-time signals","volume":"35","author":"Claasen","year":"1980","journal-title":"Philips J. Res."},{"key":"ref_23","first-page":"372","article-title":"The Wigner distribution\u2014A tool for time-frequency signal analysis\u2014Part III: Relation with other time-frequency signal transformations","volume":"35","author":"Claasen","year":"1980","journal-title":"Philips J. Res."},{"key":"ref_24","unstructured":"Papoulis, A. (1962). The Fourier Integral and Its Applications, Mc Gra-Hill Book Company, Inc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1421\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:12:29Z","timestamp":1760127149000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1421"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,15]]},"references-count":24,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["sym15071421"],"URL":"https:\/\/doi.org\/10.3390\/sym15071421","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,7,15]]}}}