{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:28:41Z","timestamp":1772252921104,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,2,15]],"date-time":"2019-02-15T00:00:00Z","timestamp":1550188800000},"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>In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution     f \u2208  S   \u2032    (  R n  )      with wavelet kernel     \u03c8 \u2208 S (  R n  )     and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of      S   \u2032    (  R n  )     . It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.<\/jats:p>","DOI":"10.3390\/sym11020235","type":"journal-article","created":{"date-parts":[[2019,2,17]],"date-time":"2019-02-17T22:11:50Z","timestamp":1550441510000},"page":"235","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Continuous Wavelet Transform of Schwartz Tempered Distributions in S\u2032 (\r\n          \r\n            \r\n              \r\n\t\t\t  \r\n                  R\r\n\t\t\t\t  n\r\n\t\t\t\t  \r\n              \r\n            \r\n          \r\n        )"],"prefix":"10.3390","volume":"11","author":[{"given":"Jagdish Narayan","family":"Pandey","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jay Singh","family":"Maurya","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi-221005, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Santosh Kumar","family":"Upadhyay","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi-221005, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","family":"Srivastava","sequence":"additional","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"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,15]]},"reference":[{"key":"ref_1","unstructured":"Bremmermann, H. (1965). Distributions, Complex Variables and Fourier Transforms, Addison-Wesley Publishing Company Inc."},{"key":"ref_2","unstructured":"Kantorovich, L.V., and Akilov, G.P. (1963). Functional Analysis in Normed Spaces, Macmillan."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Pandey, J.N. (1996). The Hilbert Transform of Schwartz Distributions and Applications, John Wiley and Sons Inc.","DOI":"10.1002\/9781118032510"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4750","DOI":"10.1090\/proc\/12590","article-title":"Continuous Wavelet transform and window functions","volume":"143","author":"Pandey","year":"2015","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_5","unstructured":"Schwartz, L. (1966). Theorie des Distributions, Hermann."},{"key":"ref_6","unstructured":"Shilov, G.E. (2005). Generalized Functions and Partial Differential Equations, Gordon and Breach Publishing Co."},{"key":"ref_7","unstructured":"Treves, F. (1997). 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Wavelets an Analysis Tool, Oxford Science Publications, Clarendon (Oxford University) Press.","DOI":"10.1093\/oso\/9780198534815.001.0001"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1515\/anly-2014-1267","article-title":"Convergence of the inverse continuous wavelet transform in Wiener amalgam spaces","volume":"35","author":"Weisz","year":"2015","journal-title":"Analysis"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s10474-012-0263-y","article-title":"Inversion Formulas for the Continuous Wavelet Transform","volume":"138","author":"Weisz","year":"2013","journal-title":"Acta Math. Hungar."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/j.amc.2016.02.013","article-title":"Computational implementation of the inverse continuous wavelet transform without a requirement of the admissibility condition","volume":"282","author":"Postnikov","year":"2016","journal-title":"Appl. Math. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/235\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:32:27Z","timestamp":1760185947000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/235"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,15]]},"references-count":21,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2]]}},"alternative-id":["sym11020235"],"URL":"https:\/\/doi.org\/10.3390\/sym11020235","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201901.0131.v1","asserted-by":"object"}]},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,2,15]]}}}