{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:43:43Z","timestamp":1760143423204,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T00:00:00Z","timestamp":1707091200000},"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":["Axioms"],"abstract":"<jats:p>We investigate L2-approximation problems in the worst case setting in the weighted Hilbert spaces H(KRd,\u03b1,\u03b3) with weights Rd,\u03b1,\u03b3 under parameters 1\u2265\u03b31\u2265\u03b32\u2265\u22ef\u22650 and 1&lt;\u03b11\u2264\u03b12\u2264\u22ef. Several interesting weighted Hilbert spaces H(KRd,\u03b1,\u03b3) appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss tractability of L2-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on d and \u03b5\u22121. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and (t1,t2)-weak tractability ((t1,t2)-WT) for all t1&gt;1 and t2&gt;0 in terms of the introduced weights under the absolute error criterion or the normalized error criterion.<\/jats:p>","DOI":"10.3390\/axioms13020108","type":"journal-article","created":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T03:28:29Z","timestamp":1707103709000},"page":"108","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces"],"prefix":"10.3390","volume":"13","author":[{"given":"Huichao","family":"Yan","sequence":"first","affiliation":[{"name":"School of Computer and Network Engineering, Shanxi Datong University, Datong 037009, China"}]},{"given":"Jia","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Berlinet, A., and Thomas-Agnan, C. (2004). Reproducing Kernel Hilbert Spaces in Probability and Statistics, Springer.","DOI":"10.1007\/978-1-4419-9096-9"},{"key":"ref_2","unstructured":"Traub, J.F., and Werschulz, A.G. (1998). Complexity and Information, Cambridge University Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Glimm, J., and Jaffe, A. (1987). Quantum Physics, Springer.","DOI":"10.1007\/978-1-4612-4728-9"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.jco.2013.05.001","article-title":"Approximation of analytic functions in Korobov spaces","volume":"30","author":"Dick","year":"2014","journal-title":"J. Complex."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/j.jat.2016.02.020","article-title":"Tractability of multivariate approximation defined over Hilbert spaces with exponential weights","volume":"207","author":"Irrgeher","year":"2016","journal-title":"J. Approx. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.jco.2017.08.002","article-title":"Average case tractability of a multivariate approximation problem","volume":"43","author":"Liu","year":"2017","journal-title":"J. Complex"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Novak, E., and Wo\u017aniakowski, H. (2008). Tractability of Multivariate Problems, Volume I: Linear Information, EMS.","DOI":"10.4171\/026"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Novak, E., and Wo\u017aniakowski, H. (2010). Tractability of Multivariate Problems, Volume II: Standard Information for Functionals, EMS.","DOI":"10.4171\/084"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Novak, E., and Wo\u017aniakowski, H. (2012). Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS.","DOI":"10.4171\/116"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"101412","DOI":"10.1016\/j.jco.2019.06.003","article-title":"A note about EC-(s, t)-weak tractability of multivariate approximation with analytic Korobov kernels","volume":"55","author":"Wang","year":"2019","journal-title":"J. Complex."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jat.2019.105309","article-title":"EC-tractability of Lp-approximation in Korobov spaces with exponential weights","volume":"249","author":"Xu","year":"2020","journal-title":"J. Approx. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"101571","DOI":"10.1016\/j.jco.2021.101571","article-title":"Tractability of approximation in the weighted Korobov space in the worst-case setting\u2014A complete picture","volume":"67","author":"Eberta","year":"2021","journal-title":"J. Complex."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s10208-002-0074-6","article-title":"Tractability of approximation for weighted Korobov spaces on classical and quantum computers","volume":"4","author":"Novak","year":"2004","journal-title":"Found. Comput. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1006\/jcom.1999.0512","article-title":"Weighted tensor product algorithms for linear multivariate problems","volume":"15","author":"Wasilkowski","year":"1999","journal-title":"J. Complex."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"101768","DOI":"10.1016\/j.jco.2023.101768","article-title":"Tractability of L2-approximation and integration in weighted Hermite spaces of finite smoothness","volume":"78","author":"Leobacher","year":"2023","journal-title":"J. Complex."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"101680","DOI":"10.1016\/j.jco.2022.101680","article-title":"EC-(t1, t2)-tractability of approximation in weighted Korobov spaces in the worst case setting","volume":"73","author":"Chen","year":"2022","journal-title":"J. Complex."},{"key":"ref_17","unstructured":"Traub, J.F., Wasilkowski, G.W., and Wo\u017aniakowski, H. (1988). Information-Based Complexity, Academic Press."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jcom.1997.0463","article-title":"When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals?","volume":"14","author":"Sloan","year":"1998","journal-title":"J. Complex."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.jco.2014.09.002","article-title":"High-dimensional integration on the Rd, weighted Hermite spaces, and orthogonal transforms","volume":"31","author":"Irrgeher","year":"2015","journal-title":"J. Complex."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"684","DOI":"10.1137\/16M1087461","article-title":"On the optimal order of integration in Hermite spaces with finite smoothness","volume":"56","author":"Dick","year":"2018","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/108\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:54:57Z","timestamp":1760104497000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/108"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,5]]},"references-count":20,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["axioms13020108"],"URL":"https:\/\/doi.org\/10.3390\/axioms13020108","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,2,5]]}}}