{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:23:35Z","timestamp":1760059415089,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,12]],"date-time":"2025-06-12T00:00:00Z","timestamp":1749686400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper investigates Abelian theorems for operators with complex Gaussian kernels over distributions of compact support. Furthermore, our investigation encompasses an examination of the Gauss\u2013Weierstrass semigroup, the linear canonical transform, and the Ornstein\u2013Uhlenbeck semigroup as particular cases within the scope of our study.<\/jats:p>","DOI":"10.3390\/axioms14060462","type":"journal-article","created":{"date-parts":[[2025,6,12]],"date-time":"2025-06-12T11:47:07Z","timestamp":1749728827000},"page":"462","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Analysis of Operators with Complex Gaussian Kernels over \u2130\u2032(\u211d): Abelian Theorems"],"prefix":"10.3390","volume":"14","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":"Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea"},{"name":"Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taoyuan City 320314, 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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5506-5050","authenticated-orcid":false,"given":"Emilio R.","family":"Negr\u00edn","sequence":"additional","affiliation":[{"name":"Departamento de An\u00e1lisis Matem\u00e1tico, Facultad de Ciencias, Universidad de La Laguna (ULL), Campus de Anchieta, ES-38271 La Laguna, Tenerife, Spain"},{"name":"Instituto de Matem\u00e1ticas y Aplicaciones (IMAULL), Universidad de La Laguna (ULL), ULL Campus de Anchieta, ES-38271 La Laguna, Tenerife, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6569-8540","authenticated-orcid":false,"given":"Jeetendrasingh","family":"Maan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Scientific Computing, National Institute of Technology, Hamirpur 177005, India"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,12]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Baez, J.C., Segal, I.E., and Zhou, Z. (1992). Introduction to Algebraic and Constructive Quantum Field Theory, Princeton University Press.","key":"ref_1","DOI":"10.1515\/9781400862504"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1090\/pspum\/048\/974332","article-title":"\u201cThe oscillator semigroup\u201d in The Mathematical Heritage of Hermann Weyl (Durham, NC, 1987)","volume":"48","author":"Howe","year":"1988","journal-title":"Proc. Sympos. Pure Math."},{"doi-asserted-by":"crossref","unstructured":"Folland, G.B. (1989). Harmonic Analysis in Phase Space, Princeton University Press. Annals of Mathematics Studies.","key":"ref_3","DOI":"10.1515\/9781400882427"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/BF01233426","article-title":"Gaussian kernels have only Gaussian maximizers","volume":"102","author":"Lieb","year":"1990","journal-title":"Invent. Math."},{"key":"ref_5","first-page":"37","article-title":"Complex Gaussian operators in dimension one","volume":"23","year":"1995","journal-title":"Bull. Inst. Math. Acad. Sin."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1185","DOI":"10.1090\/S0002-9939-1995-1227527-5","article-title":"Operators with complex Gaussian kernels: Boundedness properties","volume":"123","year":"1995","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2861","DOI":"10.2298\/FIL1909861G","article-title":"On operators with complex Gaussian kernels over Lp spaces","volume":"33","year":"2019","journal-title":"Filomat"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"833","DOI":"10.2298\/FIL2303833G","article-title":"Operators with complex Gaussian kernels: Asymptotic behaviours","volume":"37","year":"2023","journal-title":"Filomat"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1007\/s11785-016-0612-x","article-title":"Parseval-type relations and Lp-inequalities for the operators with complex Gaussian kernels","volume":"11","year":"2017","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/10652469.2011.648379","article-title":"A family of Wiener transforms associated with a pair of operators on Hilbert space","volume":"24","author":"Hayek","year":"2013","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_11","first-page":"6236","article-title":"A characterization of the second quantization by using the Segal duality transform","volume":"219","author":"Srivastava","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1255","DOI":"10.1137\/0114100","article-title":"Some Abelian theorems for the distributional Hankel and K transformations","volume":"14","author":"Zemanian","year":"1966","journal-title":"SIAM J. Appl. Math."},{"key":"ref_13","first-page":"23","article-title":"Abelian theorems for the generalized index 2F1-transform","volume":"4","author":"Hayek","year":"1992","journal-title":"Rev. Acad. Canar. Cienc."},{"key":"ref_14","first-page":"657","article-title":"The index 2F1-transform of generalized functions","volume":"34","author":"Hayek","year":"1993","journal-title":"Comment. Math. Univ. Carol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1215\/17358787-2019-0002","article-title":"Abelian theorems for distributional Kontorovich-Lebedev and Mehler-Fock transforms of general order","volume":"13","year":"2019","journal-title":"Banach J. Math. Anal."},{"key":"ref_16","first-page":"1","article-title":"Abelian theorems in the framework of the distributional index Whittaker transform","volume":"27","author":"Maan","year":"2022","journal-title":"Math. Commun."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2213","DOI":"10.1007\/s12215-022-00799-8","article-title":"Abelian theorems for Laplace, Mellin and Stieltjes transforms over distributions of compact support and generalized functions","volume":"72","author":"Maan","year":"2022","journal-title":"Rend. Circ. Mat. Palermo II Ser."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"10229","DOI":"10.2298\/FIL2330229M","article-title":"Abelian theorems for the index 2F1-transform over distributions of compact support and generalized functions","volume":"37","author":"Maan","year":"2023","journal-title":"Filomat"},{"unstructured":"Horv\u00e1th, J. (1966). Topological Vector Spaces and Distributions, Addison-Wesley.","key":"ref_19"},{"unstructured":"Schwartz, L. (1957). Th\u00e9orie des Distributions, Editions Hermann.","key":"ref_20"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1016\/j.jfa.2011.03.012","article-title":"A class of Gaussian processes with fractional spectral measures","volume":"261","author":"Alpay","year":"2011","journal-title":"J. Funct. Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1016\/0022-1236(79)90080-6","article-title":"Two-point inequalities, the Hermite semigroup, and the Gauss-Weierstrass semigroup","volume":"32","author":"Weissler","year":"1979","journal-title":"J. Funct. Anal."},{"doi-asserted-by":"crossref","unstructured":"Gonz\u00e1lez, B.J., Negr\u00edn, E.R., and Maan, J. (2024). Abelian theorems for the real Weierstrass transform over compactly supported distributions. Mathematics, 12.","key":"ref_23","DOI":"10.3390\/math12223546"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1007\/s11868-019-00275-5","article-title":"Approximation of linear canonical wavelet transform on the generalized Sobolev spaces","volume":"10","author":"Prasad","year":"2019","journal-title":"J. Pseudo-Differ. Oper. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3658","DOI":"10.1364\/JOSAA.24.003658","article-title":"Properties of the linear canonical integral transformation","volume":"24","author":"Alieva","year":"2007","journal-title":"J. Opt. Soc. Am. A"},{"doi-asserted-by":"crossref","unstructured":"Healy, J.J., Kutay, M.M., Ozaktas, H.M., and Sheridan, J.T. (2016). Linear Canonical Transform: Theory and Applications, Springer.","key":"ref_26","DOI":"10.1007\/978-1-4939-3028-9"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.sigpro.2011.07.002","article-title":"Windowed linear canonical transform and its applications","volume":"92","author":"Kou","year":"2012","journal-title":"Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.sigpro.2015.02.016","article-title":"Unified Wigner\u2013ville distribution and ambiguity function in the linear canonical transform domain","volume":"114","author":"Zhang","year":"2015","journal-title":"Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.sigpro.2015.06.010","article-title":"New Wigner distribution and ambiguity function based on the generalized translation in the linear canonical transform domain","volume":"118","author":"Zhang","year":"2016","journal-title":"Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/0022-1236(82)90103-3","article-title":"Integral transforms of analytic functions on abstract Wiener spaces","volume":"47","author":"Lee","year":"1982","journal-title":"J. Funct. Anal."},{"unstructured":"Zemanian, A.H. (1968). Generalized Integral Transformations, Interscience.","key":"ref_31"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/462\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:50:58Z","timestamp":1760032258000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/462"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,12]]},"references-count":31,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["axioms14060462"],"URL":"https:\/\/doi.org\/10.3390\/axioms14060462","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,6,12]]}}}