Quadratic spline quasi-interpolants on Powell-Sabin partitions
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Quadratic spline quasi-interpolants on Powell-Sabin partitions Carla Manni · Paul Sablonnière
Published online: 6 January 2007 © Springer Science + Business Media B.V. 2007
Abstract In this paper we address the problem of constructing quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations. Quasiinterpolants of optimal approximation order are proposed and numerical tests are presented. Keywords quasi-interpolation · quadratic splines · Powell-Sabin refinement · Bézier-Bernstein representation Mathematics Subject Classifications (2000) 65D05 · 41A05 · 41A25 · 41A50
1 Introduction It is well known that the term quasi-interpolation denotes a general approach to construct, with low computational cost, efficient local approximants to a given set of data or a given function. A quasi-interpolant (q.i.) for a given function f is usually obtained as linear combination of the elements of a suitable set of functions which are required to be positive, to ensure stability, and to have small local support in order
Communicated by: J.M. Pena. Dedicated to Prof. Mariano Gasca on the occasion of his 60th birthday. C. Manni (B) Dip. di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy e-mail: [email protected] P. Sablonnière Centre de Mathématiques, INSA de Rennes, 20 avenue des Buttes de Coësmes, CS 14315, 35043-Rennes cédex, France e-mail: [email protected]
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to achieve local control. The coefficients of the linear combination are the values of linear functionals depending on f and on its derivatives/integrals. Quasi-interpolation has received a considerable attention by many authors since the seminal paper [23]. In the univariate case various effective quasi-interpolating schemes are available, based on values of f, and/or its derivatives, and/or its integrals (see for examples [1– 3, 14–16, 23] and references quoted therein) and interesting applications have been proposed in different fields. Similarly, various interesting results have been obtained in the bivariate setting by using quasi-interpolating schemes based on tensor-product polynomial splines (see for example [1, 2, 13, 22] and references quoted therein), on spaces of splines over three directional and four directional partitions both for the uniform and the nonuniform case (see for example [4, 5, 7, 21, 22]) or on other spline spaces as those generated by simplex splines (see [5, 8, 11, 18] and references quoted therein). In this paper we address the problem of constructing quasi-interpolants in the space of quadratic splines over a Powell-Sabin refinement of a generic triangulation, [19]. The low degree and the simplicity of the Bézier-Bernstein representation coupled together with the possibility of handling arbitrary triangulations of polygonal domains, make this spline space very interesting not only from the theoretical point of view but also for applications. Thus, Powell-Sabin quadratic splines have been widely studied by severa
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