{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:21:05Z","timestamp":1760239265091,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,10,29]],"date-time":"2020-10-29T00:00:00Z","timestamp":1603929600000},"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 consider the matrix expression of convolution, and its generalized continuous form. The matrix expression of convolution is effectively applied in convolutional neural networks, and in this study, we correlate the concept of convolution in mathematics to that in convolutional neural network. Of course, convolution is a main process of deep learning, the learning method of deep neural networks, as a core technology. In addition to this, the generalized continuous form of convolution has been expressed as a new variant of Laplace-type transform that, encompasses almost all existing integral transforms. Finally, we would, in this paper, like to describe the theoretical contents as detailed as possible so that the paper may be self-contained.<\/jats:p>","DOI":"10.3390\/sym12111791","type":"journal-article","created":{"date-parts":[[2020,10,29]],"date-time":"2020-10-29T23:06:12Z","timestamp":1604012772000},"page":"1791","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Matrix Expression of Convolution and Its Generalized Continuous Form"],"prefix":"10.3390","volume":"12","author":[{"given":"Young Hee","family":"Geum","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Dankook University, Cheonan 31116, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Arjun Kumar","family":"Rathie","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Vedant College of Engineering &amp; Technology (Rajasthan Technical University), Kota 324010, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0983-164X","authenticated-orcid":false,"given":"Hwajoon","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of IT Engineering, Kyungdong University, Yangju 11458, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"LeCun, Y., Bengio, Y., and Hinton, G. (2015). Deep learning. Nature, 521.","DOI":"10.1038\/nature14539"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2017\/1762729","article-title":"The intrinsic structure and properties of Laplace-typed integral transforms","volume":"2017","author":"Kim","year":"2017","journal-title":"Math. Probl. Eng."},{"key":"ref_3","unstructured":"Kreyszig, E. (2013). Advanced Engineering Mathematics, Wiley."},{"key":"ref_4","first-page":"35","article-title":"Sumudu Transform: A new integral transform to solve differential equations and control engineering problems","volume":"24","author":"Watugula","year":"1993","journal-title":"Integr. Educ."},{"key":"ref_5","first-page":"167","article-title":"ELzaki and Sumudu Transform for Solving some Differential Equations","volume":"8","author":"Elzaki","year":"2012","journal-title":"Glob. J. Pure Appl. 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Algorithms, 81.","DOI":"10.1007\/s11075-018-0546-7"},{"key":"ref_15","unstructured":"Negnevitsky, M. (2005). Artificial Intelligence, Addison-Wesley."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Cohn, D.L. (1980). Measure Theory, Birkh\u00e4user.","DOI":"10.1007\/978-1-4899-0399-0"},{"key":"ref_17","first-page":"665","article-title":"An application of monotone convergence theorem in PDEs and Fourier analysis","volume":"98","author":"Jang","year":"2015","journal-title":"Far East J. Math. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/11\/1791\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:26:13Z","timestamp":1760178373000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/11\/1791"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,29]]},"references-count":17,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["sym12111791"],"URL":"https:\/\/doi.org\/10.3390\/sym12111791","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,10,29]]}}}