{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T08:12:00Z","timestamp":1778746320310,"version":"3.51.4"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"224","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    A general method for near-best approximations to functionals on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R Superscript d\">\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:mi>d<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , using scattered-data information is discussed. The method is actually the moving least-squares method, presented by the Backus-Gilbert approach. It is shown that the method works very well for interpolation, smoothing and derivatives\u2019 approximations. For the interpolation problem this approach gives Mclain\u2019s method. The method is near-best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation. The interpolation approximation 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 Superscript d\">\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:mi>d<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is shown to be a\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C Superscript normal infinity\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">C^\\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    function, and an approximation order result is proven for quasi-uniform sets of data points.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00974-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1517-1531","source":"Crossref","is-referenced-by-count":634,"title":["The approximation power of moving least-squares"],"prefix":"10.1090","volume":"67","author":[{"given":"David","family":"Levin","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","unstructured":"[Ab] F. Abramovici, 1984 The Shepard interpolation as the best average of a set of data, Technical Report, Tel-Aviv University."},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/BF01203417","article-title":"On quasi-interpolation by radial basis functions with scattered centres","volume":"11","author":"Buhmann, M. D.","year":"1995","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"[BG1] G. Backus and F. Gilbert, 1967 Numerical applications of a formalism for geophysical inverse problems, Geophys. J.R. Astr. Soc. 13 247-276.","DOI":"10.1111\/j.1365-246X.1967.tb02159.x"},{"key":"4","doi-asserted-by":"crossref","unstructured":"[BG2] G. Backus and F. Gilbert, 1968 The resolving power of gross Earth data, Geophys. J.R. Astr. Soc. 16 169-205.","DOI":"10.1111\/j.1365-246X.1968.tb00216.x"},{"issue":"1173","key":"5","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1098\/rsta.1970.0005","article-title":"Uniqueness in the inversion of inaccurate gross Earth data","volume":"266","author":"Backus, G.","year":"1970","journal-title":"Philos. Trans. Roy. Soc. London Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0080-4614","issn-type":"print"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1016\/0021-9045(89)90090-7","article-title":"Moving least-squares are Backus-Gilbert optimal","volume":"59","author":"Bos, L. P.","year":"1989","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":"137","DOI":"10.1093\/imanum\/10.1.137","article-title":"Data dependent triangulations for piecewise linear interpolation","volume":"10","author":"Dyn, Nira","year":"1990","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"174","key":"8","doi-asserted-by":"publisher","first-page":"577","DOI":"10.2307\/2007995","article-title":"Rate of convergence of Shepard\u2019s global interpolation formula","volume":"46","author":"Farwig, Reinhard","year":"1986","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/0377-0427(86)90175-5","article-title":"Multivariate interpolation of arbitrarily spaced data by moving least squares methods","volume":"16","author":"Farwig, Reinhard","year":"1986","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"157","key":"10","doi-asserted-by":"publisher","first-page":"181","DOI":"10.2307\/2007474","article-title":"Scattered data interpolation: tests of some methods","volume":"38","author":"Franke, Richard","year":"1982","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"11","key":"11","doi-asserted-by":"publisher","first-page":"1691","DOI":"10.1002\/nme.1620151110","article-title":"Smooth interpolation of large sets of scattered data","volume":"15","author":"Franke, Richard","year":"1980","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"issue":"155","key":"12","doi-asserted-by":"publisher","first-page":"141","DOI":"10.2307\/2007507","article-title":"Surfaces generated by moving least squares methods","volume":"37","author":"Lancaster, P.","year":"1981","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"13","doi-asserted-by":"crossref","unstructured":"[Mc1] D. H. McLain, 1974 Drawing contours from arbitrary data points, Comput. J. 17 318-324.","DOI":"10.1093\/comjnl\/17.4.318"},{"issue":"2","key":"14","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1093\/comjnl\/19.2.178","article-title":"Two dimensional interpolation from random data","volume":"19","author":"McLain, D. H.","year":"1976","journal-title":"Comput. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-4620","issn-type":"print"},{"key":"15","doi-asserted-by":"crossref","unstructured":"[Sh] D. Shepard, 1968  A two dimensional interpolation function for irregularly spaced data, Proc. 23th Nat. Conf. 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