{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:25:03Z","timestamp":1776831903441,"version":"3.51.2"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"251","license":[{"start":{"date-parts":[[2005,7,28]],"date-time":"2005-07-28T00:00:00Z","timestamp":1122508800000},"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                    In this paper, a second-order Hermite basis of the space of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">C^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -quartic splines on the six-directional mesh is constructed and the refinable mask of the basis functions is derived. In addition, the extra parameters of this basis are modified to extend the Hermite interpolating property at the integer lattices by including Lagrange interpolation at the half integers as well. We also formulate a compactly supported super function in terms of the basis functions to facilitate the construction of quasi-interpolants to achieve the highest (i.e., fifth) order of approximation in an efficient way. Due to the small (minimum) support of the basis functions, the refinable mask immediately yields (up to) four-point matrix-valued coefficient stencils of a vector subdivision scheme for efficient display of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">C^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -quartic spline surfaces. Finally, this vector subdivision approach is further modified to reduce the size of the coefficient stencils to two-point templates while maintaining the second-order Hermite interpolating property.\n                  <\/p>","DOI":"10.1090\/s0025-5718-04-01702-8","type":"journal-article","created":{"date-parts":[[2005,4,12]],"date-time":"2005-04-12T12:20:25Z","timestamp":1113308425000},"page":"1369-1390","source":"Crossref","is-referenced-by-count":7,"title":["Refinable bivariate quartic \ud835\udc36\u00b2-splines for multi-level data representation and surface display"],"prefix":"10.1090","volume":"74","author":[{"given":"Charles","family":"Chui","sequence":"first","affiliation":[]},{"given":"Qingtang","family":"Jiang","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2004,7,28]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1023\/A:1014299228104","article-title":"Smooth macro-elements based on Powell-Sabin triangle splits","volume":"16","author":"Alfeld, Peter","year":"2002","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"2","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970173","volume-title":"Multivariate splines","volume":"54","author":"Chui, Charles K.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712262"},{"key":"3","isbn-type":"print","first-page":"137","article-title":"Vertex splines and their applications to interpolation of discrete data","author":"Chui, Charles K.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0792307240"},{"key":"4","isbn-type":"print","first-page":"21","article-title":"Shape-preserving interpolation by bivariate \ud835\udc36\u00b9 quadratic splines","author":"Chui, C. K.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/9810212291"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1016\/S1063-5203(03)00062-9","article-title":"Surface subdivision schemes generated by refinable bivariate spline function vectors","volume":"15","author":"Chui, Charles K.","year":"2003","journal-title":"Appl. Comput. Harmon. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-5203","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"N. Dyn, D. Levin, J. A. Gregory, A butterfly subdivision scheme for surface interpolation with tension control, ACM Trans. Graphics 2 (1990), 160\u2013169.","DOI":"10.1145\/78956.78958"},{"key":"7","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/BF02762276","article-title":"Shift-invariant spaces and linear operator equations","volume":"103","author":"Jia, Rong-Qing","year":"1998","journal-title":"Israel J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-2172","issn-type":"print"},{"key":"8","isbn-type":"print","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1090\/amsip\/025\/13","article-title":"Approximation power of refinable vectors of functions","author":"Jia, Rong-Qing","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0821829912"},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"1071","DOI":"10.1137\/S0895479801397858","article-title":"Spectral analysis of the transition operator and its applications to smoothness analysis of wavelets","volume":"24","author":"Jia, Rong-Qing","year":"2003","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1017\/S0013091500005903","article-title":"On linear independence for integer translates of a finite number of functions","volume":"36","author":"Jia, Rong Qing","year":"1993","journal-title":"Proc. Edinburgh Math. Soc. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0013-0915","issn-type":"print"},{"key":"11","doi-asserted-by":"crossref","unstructured":"L. Kobbelt, \u221a3-subdivision, In Computer Graphics Proceedings, Annual Conference Series, 2000, pp. 103\u2013112.","DOI":"10.1145\/344779.344835"},{"key":"12","doi-asserted-by":"crossref","unstructured":"U. Labsik, G. Greiner, Interpolatory \u221a3-subdivision, Proceedings of Eurographics 2000, Computer Graphics Forum, vol. 19, 2000, pp. 131\u2013138.","DOI":"10.1111\/1467-8659.00405"},{"key":"13","unstructured":"C. Loop, Smooth subdivision surfaces based on triangles, Master\u2019s thesis, University of Utah, Department of Mathematics, Salt Lake City, 1987."},{"issue":"1-2","key":"14","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1016\/S0377-0427(00)00346-0","article-title":"Developments in bivariate spline interpolation","volume":"121","author":"N\u00fcrnberger, G.","year":"2000","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"4","key":"15","doi-asserted-by":"publisher","first-page":"316","DOI":"10.1145\/355759.355761","article-title":"Piecewise quadratic approximations on triangles","volume":"3","author":"Powell, M. J. D.","year":"1977","journal-title":"ACM Trans. Math. 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