核空间的更好基
Better bases for kernel spaces
AI总结：
研究构造稳定、局部的核空间基的可行性，提出一种使用约O(log N)^d个核组合构建基函数的方法，并证明新基是L_p稳定的且满足多项式衰减估计。
摘要：
本文研究了用核进行计算时构造稳定、局部基的可行性。特别地，我们感兴趣于构造一族$(b_ξ)_{ξ∈Ξ}$作为核空间$S(k,Ξ)$的基，使得每个基函数仅用很少的核构造。换句话说，每个函数$b_ζ(x) = \sum_{ξ∈Ξ} A_{ζ,ξ} k(x,ξ)$是核样本的线性组合，且非零系数$A_{ζ,ξ}$很少。这让人联想到从截断幂函数族构造B样条基。我们证明，对于一大类核（Sobolev核以及许多多调和及相关类型的核），这样的基存在。事实上，基元素可以使用大约$O(\log N)^d$个核的组合构造，其中$d$是流形的局部维数，$N$是核空间的维数（即$N=#Ξ$）。将此视为预处理步骤——构造基的计算成本为$O(N(\log N)^d)$。此外，我们证明新基是$L_p$稳定的，并满足相对于$Ξ$密度平稳的多项式衰减估计。
英文摘要：
In this article we investigate the feasibility of constructing stable, local bases for computing with kernels. In particular, we are interested in constructing families $(b_ξ)_{ξ\inΞ}$ that function as bases for kernel spaces $S(k,Ξ)$ so that each basis function is constructed using very few kernels. In other words, each function $b_ζ(x) = \sum_{ξ\inΞ} A_{ζ,ξ} k(x,ξ)$ is a linear combination of samples of the kernel with few nonzero coefficients $A_{ζ,ξ}$. This is reminiscent of the construction of the B-spline basis from the family of truncated power functions. We demonstrate that for a large class of kernels (the Sobolev kernels as well as many kernels of polyharmonic and related type) such bases exist. In fact, the basis elements can be constructed using a combination of roughly $O(\log N)^d$ kernels, where $d$ is the local dimension of the manifold and $N$ is the dimension of the kernel space (i.e. $N=#Ξ$). Viewing this as a preprocessing step -- the construction of the basis has computational cost $O(N(\log N)^d)$. Furthermore, we prove that the new basis is $L_p$ stable and satisfies polynomial decay estimates that are stationary with respect to the density of $Ξ$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
