{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:04:23Z","timestamp":1776791063401,"version":"3.51.2"},"reference-count":45,"publisher":"American Mathematical Society (AMS)","issue":"268","license":[{"start":{"date-parts":[[2010,1,22]],"date-time":"2010-01-22T00:00:00Z","timestamp":1264118400000},"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":"<p>\n                    Recently, a new class of surface-divergence free radial basis function interpolants has been developed for surfaces in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R cubed\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In this paper, several approximation results for this class of interpolants will be derived in the case of the sphere,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper S squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">S<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {S}^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In particular, Sobolev-type error estimates are obtained, as well as optimal stability estimates for the associated interpolation matrices. In addition, a Bernstein estimate and an inverse theorem are also derived. Numerical validation of the theoretical results is also given.\n                  <\/p>","DOI":"10.1090\/s0025-5718-09-02214-5","type":"journal-article","created":{"date-parts":[[2009,6,30]],"date-time":"2009-06-30T10:39:02Z","timestamp":1246358342000},"page":"2157-2186","source":"Crossref","is-referenced-by-count":15,"title":["Error and stability estimates for surface-divergence free RBF interpolants on the sphere"],"prefix":"10.1090","volume":"78","author":[{"given":"Edward","family":"Fuselier","sequence":"first","affiliation":[]},{"given":"Francis","family":"Narcowich","sequence":"additional","affiliation":[]},{"given":"Joseph","family":"Ward","sequence":"additional","affiliation":[]},{"given":"Grady","family":"Wright","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,1,22]]},"reference":[{"key":"1","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3658-8","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0387954511","edition":"2"},{"key":"2","doi-asserted-by":"crossref","unstructured":"T. C. Chen, Global water vapor flux and maintenance during FGGE, Mon. Wea. Rev. 113 (1985), 1801\u20131819.","DOI":"10.1175\/1520-0493(1985)113<1801:GWVFAM>2.0.CO;2"},{"issue":"1-4","key":"3","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/s11075-007-9072-8","article-title":"On choosing \u201coptimal\u201d shape parameters for RBF approximation","volume":"45","author":"Fasshauer, Gregory E.","year":"2007","journal-title":"Numer. Algorithms","ISSN":"https:\/\/id.crossref.org\/issn\/1017-1398","issn-type":"print"},{"issue":"1","key":"4","first-page":"54","article-title":"Near optimal parameter selection for multiquadric interpolation","volume":"1","author":"Foley, Thomas A.","year":"1994","journal-title":"J. Appl. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1076-5131","issn-type":"print"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1016\/j.camwa.2007.01.028","article-title":"The Runge phenomenon and spatially variable shape parameters in RBF interpolation","volume":"54","author":"Fornberg, Bengt","year":"2007","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"key":"6","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198536826.001.0001","volume-title":"Constructive approximation on the sphere","author":"Freeden, W.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0198536828"},{"key":"7","isbn-type":"print","first-page":"157","article-title":"Vector spherical spline interpolation","author":"Freeden, Willi","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/3764323841"},{"issue":"263","key":"8","doi-asserted-by":"publisher","first-page":"1407","DOI":"10.1090\/S0025-5718-07-02096-0","article-title":"Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants","volume":"77","author":"Fuselier, Edward J.","year":"2008","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1007\/s10444-008-9091-6","article-title":"Erratum: \u201cImproved stability estimates and a characterization of the native space for matrix-valued RBFs\u201d [Adv. Comput. Math. 29 (2008), no. 3, 269\u2013290; MR2438345]","volume":"29","author":"Fuselier, Edward J.","year":"2008","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"10","series-title":"Mathematics Lecture Series, No. 4","volume-title":"The index theorem and the heat equation","author":"Gilkey, Peter B.","year":"1974"},{"key":"11","unstructured":"A. E. Gill, Atmosphere-ocean dynamics, Academic Press, London, 1982."},{"key":"12","doi-asserted-by":"crossref","unstructured":"T. Gneiting, Correlation functions for atmospheric data analysis, Q. J. R. Meteorol. Soc. 125 (1999), 2449\u20132464.","DOI":"10.1002\/qj.49712555906"},{"key":"13","first-page":"117","article-title":"Optimal approximation and error bounds","author":"Golomb, Michael","year":"1959"},{"key":"14","unstructured":"J. R. Holton, An introduction to dynamic meteorology, third ed., Academic Press, San Francisco, 1992."},{"key":"15","unstructured":"S. Hubbert, Radial basis function interpolation on the sphere, Ph.D. thesis, Imperial College, 2002."},{"key":"16","series-title":"Lecture Notes in Computational Science and Engineering","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-18754-4","volume-title":"Multiresolution methods in scattered data modelling","volume":"37","author":"Iske, Armin","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/3540204792"},{"key":"17","volume-title":"Classical electrodynamics","author":"Jackson, John David","year":"1975","edition":"2"},{"issue":"226","key":"18","doi-asserted-by":"publisher","first-page":"733","DOI":"10.1090\/S0025-5718-99-01080-7","article-title":"Error estimates for scattered data interpolation on spheres","volume":"68","author":"Jetter, Kurt","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"124","DOI":"10.1016\/j.jat.2006.03.007","article-title":"Continuous and discrete least-squares approximation by radial basis functions on spheres","volume":"143","author":"Le Gia, Q. T.","year":"2006","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1016\/j.jat.2004.12.005","article-title":"Approximation in rough native spaces by shifts of smooth kernels on spheres","volume":"133","author":"Levesley, J.","year":"2005","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"key":"21","series-title":"Die Grundlehren der mathematischen Wissenschaften, Band 181","volume-title":"Non-homogeneous boundary value problems and applications. Vol. I","author":"Lions, J.-L.","year":"1972"},{"key":"22","doi-asserted-by":"crossref","unstructured":"C. L. Mader, Numerical modeling of water waves, CRC Press, Boca Raton, 2004.","DOI":"10.1201\/9780203492192"},{"issue":"4","key":"23","first-page":"77","article-title":"Multivariate interpolation and conditionally positive definite functions","volume":"4","author":"Madych, W. R.","year":"1988","journal-title":"Approx. Theory Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1000-9221","issn-type":"print"},{"key":"24","series-title":"Lecture Notes in Statistics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-7892-5","volume-title":"Spatial variation","volume":"36","author":"Mat\u00e9rn, Bertil","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/3540963650","edition":"2"},{"issue":"2","key":"25","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1006\/jath.2001.3642","article-title":"Error bounds for solving pseudodifferential equations on spheres by collocation with zonal kernels","volume":"114","author":"Morton, Tanya M.","year":"2002","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"key":"26","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0094775","volume-title":"Spherical harmonics","volume":"17","author":"M\u00fcller, Claus","year":"1966"},{"issue":"3","key":"27","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1007\/s10208-005-0197-7","article-title":"Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions","volume":"7","author":"Narcowich, Francis J.","year":"2007","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"issue":"1","key":"28","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s10444-005-7506-1","article-title":"Approximation power of RBFs and their associated SBFs: a connection","volume":"27","author":"Narcowich, Francis J.","year":"2007","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"208","key":"29","doi-asserted-by":"publisher","first-page":"661","DOI":"10.2307\/2153288","article-title":"Generalized Hermite interpolation via matrix-valued conditionally positive definite functions","volume":"63","author":"Narcowich, Francis J.","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"6","key":"30","doi-asserted-by":"publisher","first-page":"1393","DOI":"10.1137\/S0036141001395054","article-title":"Scattered data interpolation on spheres: error estimates and locally supported basis functions","volume":"33","author":"Narcowich, Francis J.","year":"2002","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"1","key":"31","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1137\/S0036141002413579","article-title":"Scattered-data interpolation on \u211d\u207f: error estimates for radial basis and band-limited functions","volume":"36","author":"Narcowich, Francis J.","year":"2004","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"4","key":"32","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1007\/s00365-002-0529-7","article-title":"Refined error estimates for radial basis function interpolation","volume":"19","author":"Narcowich, Francis J.","year":"2003","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"issue":"250","key":"33","doi-asserted-by":"publisher","first-page":"743","DOI":"10.1090\/S0025-5718-04-01708-9","article-title":"Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting","volume":"74","author":"Narcowich, Francis J.","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"34","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/s00365-005-0624-7","article-title":"Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions","volume":"24","author":"Narcowich, Francis J.","year":"2006","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"issue":"6","key":"35","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1007\/s00041-006-6903-2","article-title":"Divergence-free RBFs on surfaces","volume":"13","author":"Narcowich, Francis J.","year":"2007","journal-title":"J. Fourier Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1069-5869","issn-type":"print"},{"issue":"2-3","key":"36","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1023\/A:1018975909870","article-title":"An algorithm for selecting a good value for the parameter \ud835\udc50 in radial basis function interpolation","volume":"11","author":"Rippa, Shmuel","year":"1999","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"1","key":"37","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/BF03024331","article-title":"Distributing many points on a sphere","volume":"19","author":"Saff, E. B.","year":"1997","journal-title":"Math. Intelligencer","ISSN":"https:\/\/id.crossref.org\/issn\/0343-6993","issn-type":"print"},{"issue":"225","key":"38","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1090\/S0025-5718-99-01009-1","article-title":"Improved error bounds for scattered data interpolation by radial basis functions","volume":"68","author":"Schaback, R.","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"39","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1007\/BF02432002","article-title":"Error estimates and condition numbers for radial basis function interpolation","volume":"3","author":"Schaback, Robert","year":"1995","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"2","key":"40","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1137\/0718015","article-title":"The approximation of vector functions and their derivatives on the sphere","volume":"18","author":"Swarztrauber, Paul N.","year":"1981","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"41","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1007\/BF02123482","article-title":"Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree","volume":"4","author":"Wendland, Holger","year":"1995","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"42","series-title":"Cambridge Monographs on Applied and Computational Mathematics","isbn-type":"print","volume-title":"Scattered data approximation","volume":"17","author":"Wendland, Holger","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521843355"},{"issue":"1","key":"43","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1016\/S0021-9991(05)80016-6","article-title":"A standard test set for numerical approximations to the shallow water equations in spherical geometry","volume":"102","author":"Williamson, David L.","year":"1992","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"44","unstructured":"R. S. Womersley and I. H. Sloan, Interpolation and cubature on the sphere, accessed 2008, \\url{http:\/\/web.maths.unsw.edu.au\/ rsw\/Sphere\/}."},{"issue":"1","key":"45","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1093\/imanum\/13.1.13","article-title":"Local error estimates for radial basis function interpolation of scattered data","volume":"13","author":"Wu, Zong Min","year":"1993","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02214-5\/S0025-5718-09-02214-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02214-5\/S0025-5718-09-02214-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:12:32Z","timestamp":1776787952000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02214-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1,22]]},"references-count":45,"journal-issue":{"issue":"268","published-print":{"date-parts":[[2009,10]]}},"alternative-id":["S0025-5718-09-02214-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02214-5","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":[[2009,1,22]]}}}