{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T22:38:45Z","timestamp":1768948725509,"version":"3.49.0"},"reference-count":13,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T00:00:00Z","timestamp":1768521600000},"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>We study the Sneddon R-transform and its inverse in the setting of Lebesgue spaces. Generated by the mixed trigonometric kernel xcos(xt)+hsin(xt), the R-transform acts as a unifying operator for sine- and cosine-type integral transforms. Boundedness, continuity, and weighted Lp-estimates are established in an appropriate Banach space framework, together with Parseval\u2013Goldstein type identities. Initial and final value theorems are derived for generalized functions in Zemanian-type spaces, yielding precise asymptotic behaviour at the origin and at infinity. A finite-interval theory is also developed, leading to polynomial growth estimates and final value theorems for the finite R-transform.<\/jats:p>","DOI":"10.3390\/axioms15010063","type":"journal-article","created":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T14:46:33Z","timestamp":1768574793000},"page":"63","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Sneddon \u211b-Transform and Its Inverse over Lebesgue Spaces"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","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, AZ1007 Baku, Azerbaijan"},{"name":"Section of Mathematics, International Telematic University Uninettuno, 39 Corso Vittorio Emanuele II, 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, Spain"},{"name":"Instituto de Matem\u00e1ticas y Aplicaciones (IMAULL), Universidad de La Laguna (ULL), ULL Campus de Anchieta, ES-38271 La Laguna, 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":[[2026,1,16]]},"reference":[{"key":"ref_1","unstructured":"Sneddon, I.N. (1972). The Use of Integral Transforms, McGraw\u2013Hill."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"81","DOI":"10.36753\/mathenot.1362335","article-title":"Some Parseval-Goldstein type theorems for generalized integral transforms","volume":"12","author":"Albayrak","year":"2024","journal-title":"Math. Sci. Appl. E-Notes"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1080\/10652460701210318","article-title":"Parseval-Goldstein type identities involving the L4 and P4 transforms and their applications","volume":"18","author":"Dernek","year":"2007","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_4","first-page":"327","article-title":"Some Parseval-Goldstein type identities involving the Ls,2-transform, the Fc,2-transform and the P4-transform and their applications","volume":"202","author":"Dernek","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"33","DOI":"10.26650\/ijmath.2024.00013","article-title":"Parseval-Goldstein type theorems for integral transforms in a general setting","volume":"2","author":"Maan","year":"2024","journal-title":"Istanbul J. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1093\/imamat\/42.3.241","article-title":"Parseval-type theorem applied to certain integral transforms","volume":"42","year":"1989","journal-title":"IMA J. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s41478-017-0053-6","article-title":"Abelian theorems for the Bessel wavelet transform","volume":"28","author":"Upadhyay","year":"2020","journal-title":"J. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1750019","DOI":"10.1142\/S179355711750019X","article-title":"Abelian theorems for fractional wavelet transform","volume":"10","author":"Prasad","year":"2017","journal-title":"Asian\u2013Eur. J. Math."},{"key":"ref_9","first-page":"43","article-title":"Abelian theorems for H-transforms of functions and generalized functions","volume":"103","author":"Carmichael","year":"1987","journal-title":"J. El. Mitch. Sci. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1215\/17358787-2019-0002","article-title":"Abelian theorems for distributional Kontorovich\u2013Lebedev and Mehler\u2013Fock transforms of general order","volume":"13","year":"2019","journal-title":"Banach J. Math. Anal."},{"key":"ref_11","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_12","unstructured":"Zemanian, A.H. (1987). Generalized Integral Transformations, Dover Publications."},{"key":"ref_13","unstructured":"Horv\u00e1th, J. (1966). Topological Vector Spaces and Distributions, Addison\u2013Wesley."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/63\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T09:42:08Z","timestamp":1768902128000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/63"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,16]]},"references-count":13,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["axioms15010063"],"URL":"https:\/\/doi.org\/10.3390\/axioms15010063","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,16]]}}}