{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:56:06Z","timestamp":1776790566310,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"267","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>Interpolation by radial basis functions and interpolation by polynomials are both popular methods for function reconstruction from discrete data given on spheres. Recently, there has been an increasing interest in employing these function families together in hybrid schemes for scattered data modeling and the solution of partial differential equations on spheres. For the theoretical analysis of numerical methods for the associated discretized systems, a so-called inf-sup condition is crucial. In this paper, we derive such an inf-sup condition, and show that the constant in the inf-sup condition is independent of the polynomial degree and of the chosen point set, provided the mesh norm of the point set is sufficiently small. We then use the inf-sup condition to derive a new error analysis for the hybrid interpolation scheme of Sloan and Sommariva.<\/p>","DOI":"10.1090\/s0025-5718-09-02207-8","type":"journal-article","created":{"date-parts":[[2009,4,27]],"date-time":"2009-04-27T13:47:33Z","timestamp":1240840053000},"page":"1319-1331","source":"Crossref","is-referenced-by-count":3,"title":["Inf-sup condition for spherical polynomials and radial basis functions on spheres"],"prefix":"10.1090","volume":"78","author":[{"given":"Ian","family":"Sloan","sequence":"first","affiliation":[]},{"given":"Holger","family":"Wendland","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,1,22]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/BF01436561","article-title":"The finite element method with Lagrangian multipliers","volume":"20","author":"Babu\u0161ka, Ivo","year":"1972","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"2","first-page":"129","article-title":"On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers","volume":"8","author":"Brezzi, F.","year":"1974","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1051\/m2an\/1978120403251","article-title":"Sur l\u2019erreur d\u2019interpolation des fonctions de plusieurs variables par les \ud835\udc37^{\ud835\udc5a}-splines","volume":"12","author":"Duchon, Jean","year":"1978","journal-title":"RAIRO Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0399-0516","issn-type":"print"},{"key":"4","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","volume-title":"Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics","author":"Elman, Howard C.","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/9780198528685"},{"issue":"1","key":"5","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":"1","key":"6","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.jat.2004.04.005","article-title":"A Duchon framework for the sphere","volume":"129","author":"Hubbert, Simon","year":"2004","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1016\/j.jat.2004.04.006","article-title":"\ud835\udc3f_{\ud835\udc5d}-error estimates for radial basis function interpolation on the sphere","volume":"129","author":"Hubbert, Simon","year":"2004","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"issue":"226","key":"8","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"},{"key":"9","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":"250","key":"10","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":"11","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/s10444-007-9048-1","article-title":"Approximation on the sphere using radial basis functions plus polynomials","volume":"29","author":"Sloan, Ian H.","year":"2008","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"12","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"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-09-02207-8\/S0025-5718-09-02207-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-09-02207-8\/S0025-5718-09-02207-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:07:35Z","timestamp":1776787655000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-09-02207-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1,22]]},"references-count":12,"journal-issue":{"issue":"267","published-print":{"date-parts":[[2009,7]]}},"alternative-id":["S0025-5718-09-02207-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02207-8","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]]}}}