{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:35:36Z","timestamp":1760146536283,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,14]],"date-time":"2024-11-14T00:00:00Z","timestamp":1731542400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12001342","2022L438","2022Q10","2021-B-17","2019-B-10"],"award-info":[{"award-number":["12001342","2022L438","2022Q10","2021-B-17","2019-B-10"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Scientific and Technological Innovation Project of Colleges and Universities in Shanxi Province","award":["12001342","2022L438","2022Q10","2021-B-17","2019-B-10"],"award-info":[{"award-number":["12001342","2022L438","2022Q10","2021-B-17","2019-B-10"]}]},{"name":"Basic Youth Research Found Project of Shanxi Datong University","award":["12001342","2022L438","2022Q10","2021-B-17","2019-B-10"],"award-info":[{"award-number":["12001342","2022L438","2022Q10","2021-B-17","2019-B-10"]}]},{"name":"Doctoral Foundation Project of Shanxi Datong University","award":["12001342","2022L438","2022Q10","2021-B-17","2019-B-10"],"award-info":[{"award-number":["12001342","2022L438","2022Q10","2021-B-17","2019-B-10"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We study the multivariate L2-approximation problem APPd defined over a Banach space in the average case setting. The space is equipped with a zero-mean Gaussian measure with a weighted covariance kernel, which depends on parameter sequences \u03b1={\u03b1j}j\u2208N and \u03b2={\u03b2j}j\u2208N with 1&lt;\u03b11\u2264\u03b12\u2264\u22ef and 1\u2265\u03b21\u2265\u03b22\u2265\u22ef&gt;0. In this paper, two interesting weighted covariance kernels are considered, which model the importance of the covariance kernels. Under the absolute error criterion or the normalized error criterion, we discuss (s,t)-weak tractability of the L2-approximation problem APP={APPd}d\u2208N\u00a0from a Banach space whose zero-mean Gaussian measure has the above two weighted covariance kernels for some positive numbers s and t in the average case setting. Here, (s,t)-weak tractability means how the information complexity behaves as a function of dt and \u03b5\u2212s for large dimension d and small threshold \u03b5. In particular, for all s&gt;0 and t\u2208(0,1), we find the matching sufficient and necessary condition on the parameter sequences \u03b1={\u03b1j}j\u2208N and \u03b2={\u03b2j}j\u2208N to obtain (s,t)-weak tractability under the absolute error criterion or the normalized error criterion in the average case setting. We describe (s,t)-weak tractability by the matching sufficient and necessary condition, which reflects symmetry.<\/jats:p>","DOI":"10.3390\/sym16111527","type":"journal-article","created":{"date-parts":[[2024,11,14]],"date-time":"2024-11-14T08:06:32Z","timestamp":1731571592000},"page":"1527","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Average Case (s, t)-Weak Tractability of L2-Approximation with Weighted Covariance Kernels"],"prefix":"10.3390","volume":"16","author":[{"given":"Jia","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China"}]},{"given":"Huichao","family":"Yan","sequence":"additional","affiliation":[{"name":"School of Computer and Network Engineering, Shanxi Datong University, Datong 037009, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,14]]},"reference":[{"key":"ref_1","unstructured":"Traub, J.F., and Werschulz, A.G. 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Complex."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1527\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:32:22Z","timestamp":1760113942000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1527"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,14]]},"references-count":20,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["sym16111527"],"URL":"https:\/\/doi.org\/10.3390\/sym16111527","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,11,14]]}}}