{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T03:34:17Z","timestamp":1778297657074,"version":"3.51.4"},"reference-count":27,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,15]],"date-time":"2021-11-15T00:00:00Z","timestamp":1636934400000},"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>This study aims to achieve an efficient time-frequency representation of higher-dimensional signals by introducing the notion of a non-separable linear canonical wavelet transform in L2(Rn). The preliminary analysis encompasses the derivation of fundamental properties of the novel integral transform including the orthogonality relation, inversion formula, and the range theorem. To extend the scope of the study, we formulate several uncertainty inequalities, including the Heisenberg\u2019s, logarithmic, and Nazorav\u2019s inequalities for the proposed transform in the linear canonical domain. The obtained results are reinforced with illustrative examples.<\/jats:p>","DOI":"10.3390\/sym13112182","type":"journal-article","created":{"date-parts":[[2021,11,15]],"date-time":"2021-11-15T20:46:47Z","timestamp":1637009207000},"page":"2182","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["Non-Separable Linear Canonical Wavelet Transform"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari M.","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan"},{"name":"Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8461-869X","authenticated-orcid":false,"given":"Firdous A.","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Kashmir, Anantnag 192101, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8740-4926","authenticated-orcid":false,"given":"Tarun K.","family":"Garg","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Satyawati College, University of Delhi, Delhi 110052, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9826-9475","authenticated-orcid":false,"given":"Waseem Z.","family":"Lone","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Kashmir, Anantnag 192101, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huzaifa L.","family":"Qadri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Kashmir, Anantnag 192101, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1772","DOI":"10.1063\/1.1665805","article-title":"Linear canonical transformations and their unitary representations","volume":"12","author":"Moshinsky","year":"1971","journal-title":"J. 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Symplectic Geometry and Quantum Mechanics, Birkh\u00e4user."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Jing, R., Liu, B., Li, R., and Liu, R. (2020). The N-dimensional uncertainty principle for the free metaplectic transformation. Mathematics, 8.","DOI":"10.3390\/math8101685"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Debnath, L., and Shah, F.A. (2017). Lectuer Notes on Wavelet Transforms, Birkh\u00e4user.","DOI":"10.1007\/978-3-319-59433-0"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Debnath, L., and Shah, F.A. (2015). Wavelet Transforms and Their Applications, Birkh\u00e4user.","DOI":"10.1007\/978-0-8176-8418-1"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.cnsns.2016.06.034","article-title":"A new fractional wavelet transform","volume":"44","author":"Dai","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simulat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4491","DOI":"10.1016\/j.ijleo.2014.02.021","article-title":"Generalized wavelet transform based on the convolution operator in the linear canonical transform domain","volume":"125","author":"Wei","year":"2014","journal-title":"Optik"},{"key":"ref_12","first-page":"1","article-title":"Discrete linear canonical wavelet transform and its applications","volume":"29","author":"Wang","year":"2018","journal-title":"EURASIP J. Adv. Sig. Process."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Shah, F.A., Teali, A.A., and Tantary, A.Y. (2021). Special affine wavelet transform and the corresponding Poisson summation formula. Int. J. Wavelets Multiresolut. Inf. Process., 19.","DOI":"10.1142\/S0219691320500861"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Shah, F.A., Tantary, A.Y., and Zayed, A.I. (2020). A convolution-based special affine wavelet transforms. Integ. Trans. Special Funct., 1\u201321.","DOI":"10.1080\/10652469.2020.1844196"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1007\/s00006-021-01142-7","article-title":"Linear canonical wavelet transforms in quaternion domains","volume":"31","author":"Shah","year":"2021","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Shah, F.A., and Lone, W.Z. (2021). Quadratic-phase wavelet transform with applications to generalized differential equations. Math. Methods Appl. Sci., accepted.","DOI":"10.1002\/mma.7842"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Antoine, J.P., Murenzi, R., Vandergheynst, P., and Ali, S.T. (2004). Two-Dimensional Wavelets and Their Relatives, Cambridge University Press.","DOI":"10.1017\/CBO9780511543395"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Pandey, J.N., Pandey, J.S., Upadhyay, S.K., and Srivastava, H.M. (2019). Continuous wavelet transform of Schwartz tempered distributions in S\u2032(Rn). Symmetry, 11.","DOI":"10.20944\/preprints201901.0131.v1"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1505","DOI":"10.1007\/s11868-020-00363-x","article-title":"A family of convolution-based generalized Stockwell transforms","volume":"11","author":"Srivastava","year":"2020","journal-title":"J. Pseudo-Differ. Oper. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"11340","DOI":"10.1002\/mma.7492","article-title":"A family of Mexican hat wavelet transforms associated with an isometry in the heat equation","volume":"44","author":"Srivastava","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ali, S.T., Antoine, J.P., and Gazeau, J.P. (2014). Coherent States, Wavelets, and Their Generalizations, Springer.","DOI":"10.1007\/978-1-4614-8535-3"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1650037","DOI":"10.1142\/S0219691316500375","article-title":"The continuous wavelet transform in n-dimensions","volume":"14","author":"Pandey","year":"2016","journal-title":"Int. J. Wavelets Multiresolut. Inf. Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/BF02649110","article-title":"The uncertainty principle: A mathematical survey","volume":"3","author":"Folland","year":"1997","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1137\/0515012","article-title":"Bandwidth verses time concentration: The Heisenberg-Pauli-Weyl inequality","volume":"15","author":"Cowling","year":"1994","journal-title":"SIAM J. Math. Anal."},{"key":"ref_25","first-page":"1897","article-title":"Pitt\u2019s inequality and the uncertainty principle","volume":"123","author":"Beckner","year":"1995","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"201","DOI":"10.4171\/dm\/79","article-title":"New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform","volume":"5","author":"Wilczok","year":"2000","journal-title":"Doc. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"10416","DOI":"10.1002\/mma.7417","article-title":"Uncertainty principles for the quadratic-phase Fourier transforms","volume":"44","author":"Shah","year":"2021","journal-title":"Math. Methods Appl. 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