### Abstract

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 language | English |
---|---|

Pages (from-to) | 181-208 |

Number of pages | 28 |

Journal | Constructive Approximation |

Volume | 17 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2001 |

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### Keywords

- bivariate spline interpolation
- optimal approximation
- mathematical analysis

### Cite this

*Constructive Approximation*,

*17*(2), 181-208. https://doi.org/10.1007/s003650010034

}

*Constructive Approximation*, vol. 17, no. 2, pp. 181-208. https://doi.org/10.1007/s003650010034

**Bivariate spline interpolation with optimal approximation order.** / Davydov, Oleg; Nurnberger, G.; Zeilfelder, F.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Bivariate spline interpolation with optimal approximation order

AU - Davydov, Oleg

AU - Nurnberger, G.

AU - Zeilfelder, F.

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

KW - bivariate spline interpolation

KW - optimal approximation

KW - mathematical analysis

U2 - 10.1007/s003650010034

DO - 10.1007/s003650010034

M3 - Article

VL - 17

SP - 181

EP - 208

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

IS - 2

ER -