{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:47:51Z","timestamp":1776725271655,"version":"3.51.2"},"reference-count":26,"publisher":"American Mathematical Society (AMS)","issue":"234","license":[{"start":{"date-parts":[[2001,10,27]],"date-time":"2001-10-27T00:00:00Z","timestamp":1004140800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We show that if the open, bounded domain\n \n \n \n \n \n \u03a9\n \n <\/mml:mi>\n \n \u2282\n \n <\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n \\Omega \\subset \\mathbb {R}^{d}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has a sufficiently smooth boundary and if the data function\n \n \n \n f<\/mml:mi>\n f<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is sufficiently smooth, then the\n \n \n \n \n \n L<\/mml:mi>\n \n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n \n \u03a9\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n L_{p}(\\Omega )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -norm of the error between\n \n \n \n f<\/mml:mi>\n f<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and its surface spline interpolant is\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \n \u03b4\n \n <\/mml:mi>\n \n \n \n \u03b3\n \n <\/mml:mi>\n \n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(\\delta ^{\\gamma _{p}+1\/2})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n (\n \n \n \n \n 1<\/mml:mn>\n \n \u2264\n \n <\/mml:mo>\n p<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n 1\\leq p\\leq \\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ), where\n \n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n :=<\/mml:mo>\n min<\/mml:mo>\n {<\/mml:mo>\n m<\/mml:mi>\n ,<\/mml:mo>\n m<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n d<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n +<\/mml:mo>\n d<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n p<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n \\gamma _{p}:=\\min \\{m,m-d\/2+d\/p\\}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n m<\/mml:mi>\n m<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is an integer parameter specifying the surface spline. In case\n \n \n \n \n p<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n p=2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , this lower bound on the approximation order agrees with a previously obtained upper bound, and so we conclude that the\n \n \n \n \n L<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n L_{2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -approximation order of surface spline interpolation is\n \n \n \n \n m<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n m+1\/2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-00-01301-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:17:46Z","timestamp":1027707466000},"page":"719-737","source":"Crossref","is-referenced-by-count":8,"title":["The \ud835\udc3f\u2082-approximation order of surface spline interpolation"],"prefix":"10.1090","volume":"70","author":[{"given":"Michael","family":"Johnson","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,10,27]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"key":"2","series-title":"Van Nostrand Mathematical Studies, No. 2","volume-title":"Lectures on elliptic boundary value problems","author":"Agmon, Shmuel","year":"1965"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1006\/jath.1999.3332","article-title":"Local accuracy for radial basis function interpolation on finite uniform grids","volume":"99","author":"Bejancu, Aurelian, Jr.","year":"1999","journal-title":"J. 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