{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:56:41Z","timestamp":1759838201364,"version":"3.40.4"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2014,2,24]],"date-time":"2014-02-24T00:00:00Z","timestamp":1393200000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2014,11]]},"DOI":"10.1007\/s00211-014-0614-4","type":"journal-article","created":{"date-parts":[[2014,2,23]],"date-time":"2014-02-23T08:51:10Z","timestamp":1393145470000},"page":"589-614","source":"Crossref","is-referenced-by-count":8,"title":["Strongly elliptic pseudodifferential equations on the sphere with radial basis functions"],"prefix":"10.1007","volume":"128","author":[{"given":"T. D.","family":"Pham","sequence":"first","affiliation":[]},{"given":"T.","family":"Tran","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,2,24]]},"reference":[{"key":"614_CR1","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1090\/S0025-5718-1983-0717691-6","volume":"41","author":"DN Arnold","year":"1983","unstructured":"Arnold, D.N., Wendland, W.L.: On the asymptotic convergence of collocation methods. Math. Comp. 41, 349\u2013381 (1983)","journal-title":"Math. Comp."},{"key":"614_CR2","doi-asserted-by":"crossref","unstructured":"Babu\u0161ka, I., Banerjee, U., Osborn J.E.: Meshless and generalized finite element methods: a survey of some major results. In: Meshfree Methods for Partial Differential Equations (Bonn, 2001), volume 26 of Lecture Notes in Computational Science and Engineering, pp. 1\u201320. Springer, Berlin (2003)","DOI":"10.1007\/978-3-642-56103-0_1"},{"key":"614_CR3","doi-asserted-by":"crossref","first-page":"2733","DOI":"10.1090\/S0002-9939-03-06730-3","volume":"131","author":"D Chen","year":"2003","unstructured":"Chen, D., Menegatto, V.A., Sun, X.: A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Am. Math. Soc. 131, 2733\u20132740 (2003)","journal-title":"Proc. Am. Math. Soc."},{"key":"614_CR4","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1090\/S0025-5718-1987-0906182-9","volume":"49","author":"M Costabel","year":"1987","unstructured":"Costabel, M., Stephan, E.P.: On the convergence of collocation methods for boundary integral equations on polygons. Math. Comp. 49, 461\u2013478 (1987)","journal-title":"Math. Comp."},{"key":"614_CR5","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1090\/S0025-5718-1973-0339508-8","volume":"27","author":"J Douglas Jr","year":"1973","unstructured":"Douglas Jr, J., Dupont, T.: A finite element collocation method for quasilinear parabolic equations. Math. Comp. 27, 17\u201328 (1973)","journal-title":"Math. Comp."},{"key":"614_CR6","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198536826.001.0001","volume-title":"Constructive Approximation on the Sphere with Applications to Geomathematics","author":"W Freeden","year":"1998","unstructured":"Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere with Applications to Geomathematics. Oxford University Press, Oxford (1998)"},{"key":"614_CR7","doi-asserted-by":"crossref","unstructured":"Grafarend, E.W., Krumm, F.W., Schwarze, V.S. (eds.): Geodesy: The Challenge of the 3rd Millennium. Springer, Berlin (2003)","DOI":"10.1007\/978-3-662-05296-9"},{"key":"614_CR8","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1002\/cpa.3160180307","volume":"18","author":"L H\u00f6rmander","year":"1965","unstructured":"H\u00f6rmander, L.: Pseudodifferential operators. Comm. Pure Appl. Math. 18, 501\u2013517 (1965)","journal-title":"Comm. Pure Appl. Math."},{"key":"614_CR9","doi-asserted-by":"crossref","unstructured":"Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations, Volume 164 of Applied Mathematical Sciences. Springer, Berlin (2008)","DOI":"10.1007\/978-3-540-68545-6"},{"key":"614_CR10","first-page":"147","volume":"19","author":"E Kansa","year":"1990","unstructured":"Kansa, E.: Multiquadratics\u2014a scattered data approximation scheme with applications to computational fluid-dynamics ii: solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. 19, 147\u2013161 (1990)","journal-title":"Comput. Math."},{"key":"614_CR11","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1002\/cpa.3160180121","volume":"18","author":"J Kohn","year":"1965","unstructured":"Kohn, J., Nirenberg, L.: On the algebra of pseudodifferential operators. Comm. Pure Appl. Math. 18, 269\u2013305 (1965)","journal-title":"Comm. Pure Appl. Math."},{"key":"614_CR12","volume-title":"Strongly Elliptic Systems and Boundary Integral Equations","author":"W McLean","year":"2000","unstructured":"McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. CUP, Cambridge (2000)"},{"key":"614_CR13","doi-asserted-by":"crossref","unstructured":"Melenk, J.M.: On approximation in Meshless Methods. Frontiers of Numerical Analysis. Universitext, pp. 65\u2013141. Springer, Berlin (2005)","DOI":"10.1007\/3-540-28884-8_2"},{"key":"614_CR14","unstructured":"Morton, T.M.: Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels. PhD thesis, Graduate School, Vanderbilt University, Nashville (2000)"},{"key":"614_CR15","unstructured":"Morton, T.M.: Improved error bounds for solving pseudodifferential equations on spheres by collocation with zonal kernels. In: Trends in Approximation Theory (Nashville, TN, 2000), Innov. Appl. Math., pp. 317\u2013326. Vanderbilt Univ. Press, Nashville (2001)"},{"key":"614_CR16","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1006\/jath.2001.3642","volume":"114","author":"TM Morton","year":"2002","unstructured":"Morton, T.M., Neamtu, M.: Error bounds for solving pseudodifferential equations on spheres. J. Approx. Theory 114, 242\u2013268 (2002)","journal-title":"J. Approx. Theory"},{"key":"614_CR17","doi-asserted-by":"crossref","unstructured":"M\u00fcller, C.: Spherical Harmonics. Lecture Notes in Mathematics, vol. 17. Springer, Berlin (1966)","DOI":"10.1007\/BFb0094775"},{"key":"614_CR18","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/s10208-005-0197-7","volume":"7","author":"FJ Narcowich","year":"2007","unstructured":"Narcowich, F.J., Sun, X., Ward, J.D., Wendland, H.: Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions. Found. Comput. Math. 7, 369\u2013390 (2007)","journal-title":"Found. Comput. Math."},{"key":"614_CR19","doi-asserted-by":"crossref","first-page":"1393","DOI":"10.1137\/S0036141001395054","volume":"33","author":"FJ Narcowich","year":"2002","unstructured":"Narcowich, F.J., Ward, J.D.: Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal. 33, 1393\u20131410 (2002)","journal-title":"SIAM J. Math. Anal."},{"key":"614_CR20","volume-title":"Acoustic and Electromagnetic Equations","author":"J-C N\u00e9d\u00e9lec","year":"2000","unstructured":"N\u00e9d\u00e9lec, J.-C.: Acoustic and Electromagnetic Equations. Springer, New York (2000)"},{"key":"614_CR21","unstructured":"Petersen, B.E.: Introduction to the Fourier Transform & Pseudodifferential Operators, Volume 19 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston (1983)"},{"key":"614_CR22","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1017\/S0004972709000069","volume":"79","author":"TD Pham","year":"2009","unstructured":"Pham, T.D., Tran, T.: Solutions to pseudodifferential equations using spherical radial basis functions. Bull. Aust. Math. Soc. 79, 473\u2013485 (2009)","journal-title":"Bull. Aust. Math. Soc."},{"key":"614_CR23","unstructured":"Pham, T.D., Tran, T., Le Gia, Q.T.: Numerical solutions to a boundary-integral equation with spherical radial basis functions. In: Mercer, G.N., Roberts, A.J. (eds.) Proceedings of the 14th Biennial Computational Techniques and Applications Conference, CTAC-2008, volume 50 of ANZIAM J., pp. C266\u2013C281 (2008). http:\/\/anziamj.austms.org.au\/ojs\/index.php\/ANZIAMJ\/article\/view\/1464"},{"key":"614_CR24","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1215\/S0012-7094-42-00908-6","volume":"9","author":"IJ Schoenberg","year":"1942","unstructured":"Schoenberg, I.J.: Positive definite function on spheres. Duke Math. J. 9, 96\u2013108 (1942)","journal-title":"Duke Math. J."},{"key":"614_CR25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF03654992","volume":"8","author":"S Svensson","year":"1983","unstructured":"Svensson, S.: Pseudodifferential operators\u2014a new approach to the boundary problems of physical geodesy. Manuscr. Geod. 8, 1\u201340 (1983)","journal-title":"Manuscr. Geod."},{"key":"614_CR26","doi-asserted-by":"crossref","first-page":"2857","DOI":"10.1016\/j.apnum.2008.12.033","volume":"59","author":"T Tran","year":"2009","unstructured":"Tran, T., Le Gia, Q.T., Sloan, I.H., Stephan, E.P.: Boundary integral equations on the sphere with radial basis functions: error analysis. Appl. Numer. Math. 59, 2857\u20132871 (2009)","journal-title":"Appl. Numer. Math."},{"key":"614_CR27","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/s00211-009-0269-8","volume":"115","author":"T Tran","year":"2010","unstructured":"Tran, T., Le Gia, Q.T., Sloan, I.H., Stephan, E.P.: Preconditioners for pseudodifferential equations on the sphere with radial basis functions. Numer. Math. 115, 141\u2013163 (2010)","journal-title":"Numer. Math."},{"key":"614_CR28","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/BF02123482","volume":"4","author":"H Wendland","year":"1995","unstructured":"Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comp. Math. 4, 389\u2013396 (1995)","journal-title":"Adv. Comp. Math."},{"key":"614_CR29","doi-asserted-by":"crossref","first-page":"1521","DOI":"10.1090\/S0025-5718-99-01102-3","volume":"68","author":"H Wendland","year":"1999","unstructured":"Wendland, H.: Meshless Galerkin methods using radial basis functions. Math. Comp. 68, 1521\u20131531 (1999)","journal-title":"Math. Comp."},{"key":"614_CR30","volume-title":"Scattered Data Approximation","author":"H Wendland","year":"2005","unstructured":"Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)"},{"key":"614_CR31","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1090\/S0002-9939-1992-1096214-6","volume":"116","author":"Y Xu","year":"1992","unstructured":"Xu, Y., Cheney, E.W.: Strictly positive definite functions on spheres. Proc. Am. Math. Soc. 116, 977\u2013981 (1992)","journal-title":"Proc. Am. Math. Soc."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-014-0614-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-014-0614-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-014-0614-4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,1]],"date-time":"2025-05-01T22:32:16Z","timestamp":1746138736000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-014-0614-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,24]]},"references-count":31,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2014,11]]}},"alternative-id":["614"],"URL":"https:\/\/doi.org\/10.1007\/s00211-014-0614-4","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"type":"print","value":"0029-599X"},{"type":"electronic","value":"0945-3245"}],"subject":[],"published":{"date-parts":[[2014,2,24]]}}}