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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh


Author: Rong Qing Jia
Journal: Trans. Amer. Math. Soc. 295 (1986), 199-212
MSC: Primary 41A15; Secondary 41A25, 41A63
DOI: https://doi.org/10.1090/S0002-9947-1986-0831196-2
MathSciNet review: 831196
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Abstract: Let $ \Delta $ be the mesh in the plane obtained from a uniform square mesh by drawing in the north-east diagonal in each square. Let $ \pi _{k,\Delta }^\rho $ be the space of bivariate piecewise polynomial functions in $ {C^\rho }$, of total degree $ \leq k$, on the mesh $ \Delta $. Let $ m(k,\rho )$ denote the approximation order of $ \pi _{k,\Delta }^\rho $. In this paper, an upper bound for $ m(k,\rho )$ is given. In the space $ 3 \leq 2k - 3\rho \leq 7$, the exact values of $ m(k,\rho )$ are obtained:

\begin{displaymath}\begin{array}{*{20}{c}} {m(k,\rho ) = 2k - 2\rho - 1} \hfill ... ...or}}\;2k - 3\rho = 5,6\;{\text{or}}\;7.} \hfill \\ \end{array} \end{displaymath}

In particular, this result answers negatively a conjecture of de Boor and Höllig.

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DOI: https://doi.org/10.1090/S0002-9947-1986-0831196-2
Article copyright: © Copyright 1986 American Mathematical Society