{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:52:21Z","timestamp":1776786741159,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"260","license":[{"start":{"date-parts":[[2008,5,9]],"date-time":"2008-05-09T00:00:00Z","timestamp":1210291200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited. The conjecture that a parametric polynomial curve of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"less-than-or-equal-to n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\le n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    can interpolate\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2 n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    given points 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 squared\">\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>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is confirmed for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n less-than-or-equal-to 5\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>5<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n \\le 5<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2 n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    can be achieved as soon as the interpolating curve exists.\n                  <\/p>","DOI":"10.1090\/s0025-5718-07-01988-6","type":"journal-article","created":{"date-parts":[[2007,7,26]],"date-time":"2007-07-26T07:46:03Z","timestamp":1185435963000},"page":"1981-1993","source":"Crossref","is-referenced-by-count":20,"title":["On geometric interpolation by planar parametric polynomial curves"],"prefix":"10.1090","volume":"76","author":[{"given":"Ga\u0161per","family":"Jakli\u010d","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jernej","family":"Kozak","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marjeta","family":"Krajnc","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emil","family":"\u017dagar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2007,5,9]]},"reference":[{"key":"1","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61257-2","volume-title":"Numerical continuation methods","volume":"13","author":"Allgower, Eugene L.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3540127607"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1016\/0167-8396(87)90002-1","article-title":"High accuracy geometric Hermite interpolation","volume":"4","author":"de Boor, Carl","year":"1987","journal-title":"Comput. Aided Geom. Design","ISSN":"https:\/\/id.crossref.org\/issn\/0167-8396","issn-type":"print"},{"key":"3","isbn-type":"print","first-page":"167","article-title":"On spline interpolation of space data","author":"Feng, Yu Yu","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0826513158"},{"issue":"8","key":"4","doi-asserted-by":"publisher","first-page":"681","DOI":"10.1016\/0167-8396(96)00004-0","article-title":"Geometric Hermite interpolation with maximal order and smoothness","volume":"13","author":"H\u00f6llig, K.","year":"1996","journal-title":"Comput. Aided Geom. Design","ISSN":"https:\/\/id.crossref.org\/issn\/0167-8396","issn-type":"print"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"953","DOI":"10.1137\/S0036142903422077","article-title":"On geometric interpolation by polynomial curves","volume":"42","author":"Kozak, Jernej","year":"2004","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"6","first-page":"Art. B42q, 67","article-title":"Advanced determinant calculus","volume":"42","author":"Krattenthaler, C.","year":"1999","journal-title":"S\\'{e}m. Lothar. Combin."},{"key":"7","isbn-type":"print","first-page":"385","article-title":"Parametric interpolation by quadratic polynomials in the plane","author":"M\u00f8rken, Knut","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0826512682"},{"issue":"217","key":"8","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1090\/S0025-5718-97-00796-5","article-title":"A general framework for high-accuracy parametric interpolation","volume":"66","author":"M\u00f8rken, Knut","year":"1997","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":"89","DOI":"10.1016\/0167-8396(94)00004-C","article-title":"High order approximation method for curves","volume":"12","author":"Rababah, Abedallah","year":"1995","journal-title":"Comput. Aided Geom. Design","ISSN":"https:\/\/id.crossref.org\/issn\/0167-8396","issn-type":"print"},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1016\/0167-8396(89)90025-3","article-title":"Interpolation with piecewise quadratic visually \ud835\udc36\u00b2 B\u00e9zier polynomials","volume":"6","author":"Schaback, Robert","year":"1989","journal-title":"Comput. Aided Geom. Design","ISSN":"https:\/\/id.crossref.org\/issn\/0167-8396","issn-type":"print"},{"key":"11","unstructured":"Karl Scherer, Parametric polynomial curves of local approximation order 8, Curve and Surface Fitting (Saint Malo, 1999), Vanderbilt Univ. Press, Nashville, TN, 2000, pp. 375\u2013384."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2007-76-260\/S0025-5718-07-01988-6\/S0025-5718-07-01988-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2007-76-260\/S0025-5718-07-01988-6\/S0025-5718-07-01988-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:03:26Z","timestamp":1776783806000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2007-76-260\/S0025-5718-07-01988-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,5,9]]},"references-count":11,"journal-issue":{"issue":"260","published-print":{"date-parts":[[2007,10]]}},"alternative-id":["S0025-5718-07-01988-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-07-01988-6","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":[[2007,5,9]]}}}