Approximation of Functions of Few Variables in High Dimensions
Approximation of Functions of Few Variables in High Dimensions
复制标题
高维少变量函数的逼近
DOI：
10.1007/s00365-010-9105-8
复制
发表时间：
2011
影响因子：
2.7
通讯作者：
P. Wojtaszczyk
中科院分区：
文献类型：
--
作者：
R. DeVore;G. Petrova;P. Wojtaszczyk
设f是定义在Ω上的连续函数：=[0，1]N，它只依赖于ℓ坐标变量，$f(x_{1}，\ldots，x_{N})=g(x_{i_{1}，\ldots，x_{i_{ell}})$。我们假设我们被给予m，并且被允许在Ω中的m个点处询问f的值。如果g在Lip1中，并且坐标i1，…，iℓ是已知的，则通过求m=Lℓ等距点处f的值，我们可以将f恢复到C(−)范数下的精度|g|Lip1LΩ1。本文研究了当坐标i1、…，Iℓ不为我们所知。本文的一个典型结果是，通过要求C(ℓ)Lℓ(LOG 2N)自适应地选择f的点值，我们可以在一致范数下将f恢复到当g−1Lip1时的精度|g|∈1L Lip1。对于g上更一般的光滑性条件，也证明了类似的结果。在假设f可以用ε变量的函数逼近于某个容差ℓ(未知)的情况下，也证明了这些结果。
Let f be a continuous function defined on Ω:=[0,1]N which depends on only ℓ coordinate variables, $f(x_{1},\ldots,x_{N})=g(x_{i_{1}},\ldots,x_{i_{\ell}})$. We assume that we are given m and are allowed to ask for the values of f at m points in Ω. If g is in Lip1 and the coordinates i1,…,iℓ are known to us, then by asking for the values of f at m=Lℓ uniformly spaced points, we could recover f to the accuracy |g|Lip1L−1 in the norm of C(Ω). This paper studies whether we can obtain similar results when the coordinates i1,…,iℓ are not known to us. A prototypical result of this paper is that by asking for C(ℓ)Lℓ(log 2N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy |g|Lip1L−1 when g∈Lip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance ε (which is not known) by functions of ℓ variables.