{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:42:31Z","timestamp":1760146951600,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T00:00:00Z","timestamp":1735603200000},"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":["11671213"],"award-info":[{"award-number":["11671213"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We introduce the H\u00f6lder width, which measures the best error performance of some recent nonlinear approximation methods, such as deep neural network approximation. Then, we investigate the relationship between H\u00f6lder widths and other widths, showing that some H\u00f6lder widths are essentially smaller than n-Kolmogorov widths and linear widths. We also prove that, as the H\u00f6lder constants grow with n, the H\u00f6lder widths are much smaller than the entropy numbers. The fact that H\u00f6lder widths are smaller than the known widths implies that the nonlinear approximation represented by deep neural networks can provide a better approximation order than other existing approximation methods, such as adaptive finite elements and n-term wavelet approximation. In particular, we show that H\u00f6lder widths for Sobolev and Besov classes, induced by deep neural networks, are O(n\u22122s\/d) and are much smaller than other known widths and entropy numbers, which are O(n\u2212s\/d).<\/jats:p>","DOI":"10.3390\/axioms14010025","type":"journal-article","created":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T10:17:40Z","timestamp":1735640260000},"page":"25","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Theory and Applications of H\u00f6lder Widths"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-6521-7768","authenticated-orcid":false,"given":"Man","family":"Lu","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4706-3223","authenticated-orcid":false,"given":"Peixin","family":"Ye","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"107","DOI":"10.2307\/1968691","article-title":"Uber die beste Annaherung von Funktionen einer gegebenen Funktionenklasse","volume":"37","author":"Kolmogoroff","year":"1936","journal-title":"Ann. 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