{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T23:10:59Z","timestamp":1787008259967,"version":"build-2736575974"},"reference-count":35,"publisher":"American Mathematical Society (AMS)","issue":"359","license":[{"start":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T00:00:00Z","timestamp":1775692800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["EXC 2075 - 390740016"],"award-info":[{"award-number":["EXC 2075 - 390740016"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["05M20VSA"],"award-info":[{"award-number":["05M20VSA"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002347","name":"Bundesministerium f\u00fcr Bildung und Forschung","doi-asserted-by":"publisher","award":["EXC 2075 - 390740016"],"award-info":[{"award-number":["EXC 2075 - 390740016"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002347","name":"Bundesministerium f\u00fcr Bildung und Forschung","doi-asserted-by":"publisher","award":["05M20VSA"],"award-info":[{"award-number":["05M20VSA"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    While direct statements for kernel based interpolation on regions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega subset-of double-struck upper R Superscript d\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u03a9\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2282\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>d<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Omega \\subset \\mathbb {R}^d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are well researched, far less is known about corresponding inverse statements. The available inverse statements for kernel based interpolation so far are not sharp.\n                  <\/p>\n                  <p>In this paper, we derive sharp inverse statements for interpolation using finitely smooth kernels, such as popular radial basis function kernels like the class of Mat\u00e9rn or Wendland kernels. In particular, the results show that there is a one-to-one correspondence between the smoothness of a function and its approximation rate via kernel interpolation: If a function can be approximated with a given rate, it has a corresponding smoothness and vice versa.<\/p>","DOI":"10.1090\/mcom\/4094","type":"journal-article","created":{"date-parts":[[2025,3,26]],"date-time":"2025-03-26T14:19:22Z","timestamp":1742998762000},"page":"1389-1413","source":"Crossref","is-referenced-by-count":7,"title":["Sharp inverse statements for kernel interpolation"],"prefix":"10.1090","volume":"95","author":[{"given":"Tizian","family":"Wenzel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,4,9]]},"reference":[{"key":"1","series-title":"Springer Monographs in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-14648-5","volume-title":"Sobolev spaces, their generalizations and elliptic problems in smooth and Lipschitz domains","author":"Agranovich, Mikhail S.","year":"2015","ISBN":"https:\/\/id.crossref.org\/isbn\/9783319146478"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1007\/s00211-007-0092-z","article-title":"An extension of a bound for functions in Sobolev spaces, with applications to (\ud835\udc5a,\ud835\udc60)-spline interpolation and smoothing","volume":"107","author":"Arcang\u00e9li, R\u00e9mi","year":"2007","journal-title":"Numer. 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