arXiv math/04104882004-10-22math.NAcs.NA

近最佳样条拟插值的最新结果

Recent Results on Near-Best Spline Quasi-Interpolants

Paul Sablonniere

AI总结：

本文综述了近最佳样条拟插值算子的构造方法，通过最小化系数向量的1-范数来获得无穷范数的上界，并给出了均匀和非均匀剖分下的一元和二元拟插值的新结果。

摘要：

粗略地说，近最佳（简称NB）拟插值（简称QI）是一种形如$Q_af=\sum_{\alpha\in A} \Lambda_\alpha(f) B_\alpha$的逼近算子，其中$B_\alpha$是B样条，$\Lambda_\alpha(f)$是作用于给定函数$f$的线性离散或积分形式。这些形式依赖于有限个系数，这些系数是向量$a_\alpha$的分量，$\alpha\in A$。指标$a$指代这一向量序列。为了使$Q_a p=p$对所有属于由$B_\alpha$生成的样条空间中的某个子空间的多项式$p$成立，每个向量$a_\alpha$必须位于一个仿射子空间$V_\alpha$中，即满足一些线性约束。然而，仍存在一些自由度，用于对每个$\alpha\in A$最小化$\Vert a_\alpha\Vert_1$。容易证明$\max \{\Vert a_\alpha\Vert_1 ; \alpha\in A\}$是$\Vert Q_a \Vert_{\infty}$的一个上界：因此，我们不直接最小化$Q_a$的无穷范数（这是一个困难问题），而是最小化该范数的一个上界，这要容易得多。此外，后一个问题总是至少有一个解，该解对应于一个NB QI。在论文的第一部分，我们综述了作者及其合作者已研究的定义在均匀或非均匀剖分上的NB一元或二元样条QI。在第二部分，我们给出了一些新结果，主要关于非均匀剖分上的一元和二元积分QI：在这种情况下，NB QI更难刻画，最优性质强烈依赖于剖分的几何形状。因此，我们将研究限制在具有有趣形状性质和/或无穷范数一致有界（与剖分无关）的QI上。

英文摘要：

Roughly speaking, a near-best (abbr. NB) quasi-interpolant (abbr. QI) is an approximation operator of the form $Q_af=\sum_{α\in A} Λ_α(f) B_α$ where the $B_α$'s are B-splines and the $Λ_α(f)$'s are linear discrete or integral forms acting on the given function $f$. These forms depend on a finite number of coefficients which are the components of vectors $a_α$ for $α\in A$. The index $a$ refers to this sequence of vectors. In order that $Q_a p=p$ for all polynomials $p$ belonging to some subspace included in the space of splines generated by the $B_α$'s, each vector $a_α$ must lie in an affine subspace $V_α$, i.e. satisfy some linear constraints. However there remain some degrees of freedom which are used to minimize $\Vert a_α\Vert_1$ for each $α\in A$. It is easy to prove that $\max \{\Vert a_α\Vert_1 ; α\in A\}$ is an upper bound of $\Vert Q_a \Vert_{\infty}$: thus, instead of minimizing the infinite norm of $Q_a$, which is a difficult problem, we minimize an upper bound of this norm, which is much easier to do. Moreover, the latter problem has always at least one solution, which is associated with a NB QI. In the first part of the paper, we give a survey on NB univariate or bivariate spline QIs defined on uniform or non-uniform partitions and already studied by the author and coworkers. In the second part, we give some new results, mainly on univariate and bivariate integral QIs on {\sl non-uniform} partitions: in that case, NB QIs are more difficult to characterize and the optimal properties strongly depend on the geometry of the partition. Therefore we have restricted our study to QIs having interesting shape properties and/or infinite norms uniformly bounded independently of the partition.

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