Bivariate spline interpolation with optimal approximation order

Oleg Davydov, G. Nurnberger, F. Zeilfelder

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21 Citations (Scopus)
9 Downloads (Pure)


Let be a triangulation of some polygonal domain f c R2 and let S9 (A)
denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to A. We develop the first Hermite-type interpolation scheme for S9 (A), q >_ 3r + 2, whose approximation error is bounded above by Kh4+i, where h is the maximal diameter of the triangles in A, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and nearsingular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of Sr, (A). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [71 and [181.
Original languageEnglish
Pages (from-to)181-208
Number of pages28
JournalConstructive Approximation
Issue number2
Publication statusPublished - 2001


  • bivariate spline interpolation
  • optimal approximation
  • mathematical analysis


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