{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T18:42:34Z","timestamp":1767984154360,"version":"3.49.0"},"reference-count":23,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,22]],"date-time":"2024-07-22T00:00:00Z","timestamp":1721606400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Yuxiu Innovation Project of NCUT","award":["2024NCUTYXCX104"],"award-info":[{"award-number":["2024NCUTYXCX104"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we mainly studied the Gel\u2019fand widths of Sobolev space in the probabilistic and average settings. And, we estimated the sharp bounds of the probabilistic Gel\u2019fand (N,\u03b4)-widths of multivariate Sobolev space MW2r(Td) with mixed derivative equipped with the Gaussian measure in the Sq-norm by discretization methods. Later, we estimated the sharp bounds of the p-average Gel\u2019fand N-widths of univariate Sobolev space W2r(T) and multivariate Sobolev space MW2r(Td) with mixed derivative equipped with the Gaussian measure in the Sq-norm.<\/jats:p>","DOI":"10.3390\/axioms13070492","type":"journal-article","created":{"date-parts":[[2024,7,22]],"date-time":"2024-07-22T17:36:04Z","timestamp":1721669764000},"page":"492","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Probabilistic and Average Gel\u2019fand Widths of Sobolev Space Equipped with Gaussian Measure in the Sq-Norm"],"prefix":"10.3390","volume":"13","author":[{"given":"Ruihuan","family":"Wu","sequence":"first","affiliation":[{"name":"College of Science, North China University of Technology, Beijing 100144, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5830-8417","authenticated-orcid":false,"given":"Yuqi","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Science, North China University of Technology, Beijing 100144, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huan","family":"Li","sequence":"additional","affiliation":[{"name":"College of Science, North China University of Technology, Beijing 100144, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Pinkus, A. 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