论文标题
点式最佳多元样条方法，用于恢复单纯函数两倍的函数
Pointwise optimal multivariate spline method for recovery of twice differentiable functions on a simplex
论文作者
论文摘要
我们在$ t $的顶点上用作函数$ f $的信息值和梯度，在$ t $的顶点使用$ d $二维的单纯型$ t $在$ t $ $ f（{\ bf w}）$的恢复时，每个点$ {\ bf w}）$在$ {\ bf bf w} $上的$ f $ p $ p $上的$ p $ p $ a $ t $ w^2（在任何方向上均匀边界的二阶导数。如果特别是（任何维度）的$ t $的每张面都包含其圆周，我们可以服用$ p = t $。 我们还找到了点式最佳方法的错误函数，该方法被证明是$ W^2（p）$中的函数，并具有零信息。错误函数是在某个多面体分区上的分段二次$ c^1 $ function，可以被视为经典Euler spline $ ϕ_2 $的多元类似物。最佳方法是在同一分区上的第二度（具有一定程度）的连续样条。
We obtain the spline recovery method on a $d$-dimensional simplex $T$ that uses as information values and gradients of a function $f$ at the vertices of $T$ and is optimal for recovery of $f({\bf w})$ at every point ${\bf w}$ of an admissible domain $P$ containing $T$ on the class $W^2(P)$ of twice differentiable functions on $P$ with uniformly bounded second order derivatives in any direction. If, in particular, every face of $T$ (of any dimension) contains its circumcenter, we can take $P=T$. We also find the error function of the pointwise optimal method which turns out to be a function in $W^2(P)$ with zero information. The error function is a piecewise quadratic $C^1$-function over a certain polyhedral partition and can be considered as a multivariate analogue of the classical Euler spline $ϕ_2$. The pointwise optimal method is a continuous spline of degree two (with some pieces of degree one) over the same partition.
