Least-squares approximation by second-order L-splines
Alexander Kayumov
备注：
5 pages, in French. These results were presented at the Eleventh International Conference in Approximation Theory (Gatlinburg, Tennessee, USA, May 18-22, 2004) and are contained in the author's Master's thesis defended in 2003 at the Ural State University (Ekaterinburg, Russia). This is a translation of the first chapter of the thesis into French
在1973年的一篇论文中，Carl de Boor提出了一个猜想，26年后的1999年，Alexei Shadrin完全证明了对于任意阶的样条空间，样条正交投影算子的$L_\infty$范数独立于节点序列有界。在Shadrin的证明出现之后，很自然地将正交投影算子的$L_\infty$有界性研究推广到L样条。本文是这方面的一个初步尝试。 我们考虑由某些具有常系数的二阶线性微分算子生成的L样条空间上的正交投影算子。我们关注这些正交投影算子的$L_2$范数是否独立于节点序列有界。 对于由算子$L=D^2-\alpha^2$生成的指数样条，我们找到了精确的与网格无关的界，该界与对应的线性样条多项式情况相同，即3。对于由算子$L=(D-\alpha^2)(D+\beta^2)$生成的指数样条，我们证明了与网格无关的有界性，但没有给出显式界。 对于由算子$L=D^2+\alpha^2$生成的三角样条，在分区直径的标准约束下，我们找到了一个与网格无关的界。我们还考虑了一个模型例子，该例子表明上述约束对于实现与网格无关的有界性并非必要——由于一种有趣的“干涉”现象，$L_2$逼近保持有界，而它的系数和LB样条的范数都趋于无穷。
英文摘要：
In a 1973 paper Carl de Boor conjectured and 26 years later in 1999 Alexei Shadrin proved in full generality that the $L_\infty$-norm of the spline orthoprojector is bounded independently of the knot sequence for every order of the spline space. Following the appearance of Shadrin's proof, it was only very natural to carry the study of $L_\infty$-boundedness of the orthoprojector over to L-splines. This is a modest attempt in the direction.
We consider orthoprojectors on spaces of L-splines generated by certain second-order linear differential operators with constant coefficients. We concern ourselves with whether the $L_2$-norms of the resulting orthoprojectors are bounded independenly of the knot sequence.
For exponential splines generated by the operator $L=D^2-α^2$ we find the exact mesh-independent bound, which proves to be the same as in the corresponding polynomial case of linear splines, that is 3. For exponential splines generated by the operator $L=(D-α^2)(D+β^2)$ we prove mesh-independent boundedness without finding an explicit bound.
For trigonometric splines generated by the operator $L=D^2+α^2$ with standard constraints on the partition diameter we find a mesh-independent bound. We also consider a model example which shows that the above constraints are not necessary for mesh-independent boundedness to hold - due to an interesting "interference" phenomenon, the $L_2$-approximant remains bounded while both its coefficients and the norms of LB-splines tend to infinity.
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