{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T11:29:38Z","timestamp":1777462178948,"version":"3.51.4"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"252","license":[{"start":{"date-parts":[[2006,2,14]],"date-time":"2006-02-14T00:00:00Z","timestamp":1139875200000},"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 Radial Basis Functions (RBF) have found a wide area of applications. We consider the case of polyharmonic RBF (called sometimes polyharmonic splines) where the data are on special grids of the form\n \n \n \n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n \u00d7\n \n <\/mml:mo>\n a<\/mml:mi>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n \\mathbb {Z}\\times a\\mathbb {Z}^{n}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n having practical importance. The main purpose of the paper is to consider the behavior of the polyharmonic interpolation splines\n \n \n \n \n I<\/mml:mi>\n \n a<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n I_{a}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n on such grids for the limiting process\n \n \n \n \n a<\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n <\/mml:mrow>\n a\\rightarrow 0,<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n \n \n \n \n a<\/mml:mi>\n ><\/mml:mo>\n 0.<\/mml:mn>\n <\/mml:mrow>\n a>0.<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n For a large class of data functions defined on\n \n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n \u00d7\n \n <\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n \\mathbb {R}\\times \\mathbb {R}^{n}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n it turns out that there exists a limit function\n \n \n \n \n I<\/mml:mi>\n .<\/mml:mo>\n <\/mml:mrow>\n I.<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n This limit function is shown to be a\n polyspline<\/italic>\n of order\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n on strips. By the theory of polysplines we know that the function\n \n \n \n I<\/mml:mi>\n I<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is smooth up to order\n \n \n \n \n 2<\/mml:mn>\n \n (<\/mml:mo>\n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n 2\\left ( p-1\\right )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n everywhere (in particular, they are smooth on the hyperplanes\n \n \n \n \n \n {<\/mml:mo>\n j<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n \n \u00d7\n \n <\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n \\left \\{ j\\right \\} \\times \\mathbb {R}^{n}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , which includes existence of the normal derivatives up to order\n \n \n \n \n 2<\/mml:mn>\n \n (<\/mml:mo>\n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n 2\\left ( p-1\\right ))<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n while the RBF interpolants\n \n \n \n \n I<\/mml:mi>\n \n a<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n I_{a}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are smooth only up to the order\n \n \n \n \n 2<\/mml:mn>\n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n n<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1.<\/mml:mn>\n <\/mml:mrow>\n 2p-n-1.<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n The last fact has important consequences for the data smoothing practice.\n <\/p>","DOI":"10.1090\/s0025-5718-05-01753-9","type":"journal-article","created":{"date-parts":[[2005,8,10]],"date-time":"2005-08-10T10:23:21Z","timestamp":1123669401000},"page":"1831-1841","source":"Crossref","is-referenced-by-count":10,"title":["Polyharmonic splines on grids \u2124\u00d7\ud835\udd52\u2124\u207f and their limits"],"prefix":"10.1090","volume":"74","author":[{"given":"O.","family":"Kounchev","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"H.","family":"Render","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2005,2,14]]},"reference":[{"key":"1","isbn-type":"print","first-page":"41","article-title":"Cardinal interpolation with biharmonic polysplines on strips","author":"Bejancu, Aurelian","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0972848215"},{"key":"2","unstructured":"Bejancu, A., Kounchev, O., Render, H., The cardinal interpolation on hyperplanes with polysplines, submitted."},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/BF01890410","article-title":"Multivariate cardinal interpolation with radial-basis functions","volume":"6","author":"Buhmann, M. D.","year":"1990","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1017\/S0305004100071000","article-title":"On radial basis approximation on periodic grids","volume":"112","author":"Buhmann, Martin D.","year":"1992","journal-title":"Math. Proc. Cambridge Philos. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0305-0041","issn-type":"print"},{"key":"5","series-title":"Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-96750-4","volume-title":"The analysis of linear partial differential operators. I","volume":"256","author":"H\u00f6rmander, Lars","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/3540121048"},{"key":"6","isbn-type":"print","first-page":"131","article-title":"Multivariate approximation from the cardinal interpolation point of view","author":"Jetter, K.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0121745899"},{"key":"7","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1198\/tech.2001.s63","volume-title":"Multivariate polysplines: applications to numerical and wavelet analysis","author":"Kounchev, Ognyan","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0124224903"},{"key":"8","unstructured":"Kounchev, O., Render, H., Multivariate cardinal splines via spherical harmonics, submitted"},{"key":"9","isbn-type":"print","first-page":"333","article-title":"Wavelet analysis of cardinal \ud835\udc3f-splines and construction of multivariate prewavelets","author":"Kounchev, Ognyan","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0826514162"},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1090\/S0002-9939-03-07069-2","article-title":"The approximation order of polysplines","volume":"132","author":"Kounchev, Ognyan","year":"2004","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"11","unstructured":"Kounchev, O., Render, H., Rate of convergence of polyharmonic splines to polysplines. Submitted."},{"key":"12","doi-asserted-by":"crossref","unstructured":"Kounchev, O., Wilson, M., Application of PDE methods to visualization of heart data. In: Michael J. Wilson, Ralph R. Martin (Eds.): Mathematics of Surfaces, Lecture Notes in Computer Science 2768, Springer-Verlag, 2003; pp. 377-391.","DOI":"10.1007\/978-3-540-39422-8_25"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1006\/jath.1999.3360","article-title":"Simultaneous approximations for functions in Sobolev spaces by derivatives of polyharmonic cardinal splines","volume":"101","author":"Liu, Yongping","year":"1999","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"issue":"2","key":"14","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/0021-9045(90)90079-6","article-title":"Polyharmonic cardinal splines","volume":"60","author":"Madych, W. R.","year":"1990","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"issue":"189","key":"15","doi-asserted-by":"publisher","first-page":"211","DOI":"10.2307\/2008691","article-title":"Multivariate interpolation and conditionally positive definite functions. II","volume":"54","author":"Madych, W. R.","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"16","series-title":"Princeton Mathematical Series, No. 32","volume-title":"Introduction to Fourier analysis on Euclidean spaces","author":"Stein, Elias M.","year":"1971"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2005-74-252\/S0025-5718-05-01753-9\/S0025-5718-05-01753-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2005-74-252\/S0025-5718-05-01753-9\/S0025-5718-05-01753-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:30:17Z","timestamp":1776781817000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2005-74-252\/S0025-5718-05-01753-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,2,14]]},"references-count":16,"journal-issue":{"issue":"252","published-print":{"date-parts":[[2005,10]]}},"alternative-id":["S0025-5718-05-01753-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-05-01753-9","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2005,2,14]]}}}