$L_\infty$-逼近在具有指数权重的Korobov空间中
$\boldsymbol{L}_{\infty}$-approximation in Korobov spaces with Exponential Weights
AI总结：
研究了加权Korobov空间中周期函数的多变量$\boldsymbol{L}_{\infty}$逼近问题，探讨了误差收敛性和不同维度下的可处理性 性质。
摘要：
我们研究了加权Korobov空间中周期函数的多变量$\boldsymbol{L}_{\infty}$逼近问题，其中傅里叶系数以指数速度衰减。权重由两个正实数序列$\boldsymbol{a}=\{a_j\}$和$\boldsymbol{b}=\{b_j\}$定义。我们研究了使用$n$次信息评估从类$Λ$中所有算法的最小最坏误差$e^{\boldsymbol{L}_{\infty}\mathrm{-app},Λ}(n,s)$。我们考虑了两种类$Λ$：所有线性函数的类$Λ^{\rm all}$和仅函数评估的类$Λ^{\rm std}$。我们研究了最小最坏误差的指数收敛性，即随着$n$增加，误差以指数速度趋于零。此外，我们探讨了误差如何依赖于维度$s$。为此，我们定义了$\kappa$-EC-weak、EC-多项式和EC-强多项式 tractability 的概念，其中EC代表“指数收敛”。特别是，EC-多项式 tractability 意味着需要多项式数量的信息评估在$s$和$1+\log\,\varepsilon^{-1}$中计算一个$\varepsilon$-近似。我们推导了序列$\boldsymbol{a}$和$\boldsymbol{b}$的必要和充分条件，以获得指数误差收敛性和各种 tractability 性质。结果对于两种类$Λ$都是相同的。
英文摘要：
We study multivariate $\boldsymbol{L}_{\infty}$-approximation for a weighted Korobov space of periodic functions for which the Fourier coefficients decay exponentially fast. The weights are defined, in particular, in terms of two sequences $\boldsymbol{a}=\{a_j\}$ and $\boldsymbol{b}=\{b_j\}$ of positive real numbers bounded away from zero. We study the minimal worst-case error $e^{\boldsymbol{L}_{\infty}\mathrm{-app},Λ}(n,s)$ of all algorithms that use $n$ information evaluations from a class $Λ$ in the $s$-variate case. We consider two classes $Λ$ in this paper: the class $Λ^{\rm all}$ of all linear functionals and the class $Λ^{\rm std}$ of only function evaluations. We study exponential convergence of the minimal worst-case error, which means that $e^{\boldsymbol{L}_{\infty}\mathrm{-app},Λ}(n,s)$ converges to zero exponentially fast with increasing $n$. Furthermore, we consider how the error depends on the dimension $s$. To this end, we define the notions of $κ$-EC-weak, EC-polynomial and EC-strong polynomial tractability, where EC stands for "exponential convergence". In particular, EC-polynomial tractability means that we need a polynomial number of information evaluations in $s$ and $1+\log\,\varepsilon^{-1}$ to compute an $\varepsilon$-approximation. We derive necessary and sufficient conditions on the sequences $\boldsymbol{a}$ and $\boldsymbol{b}$ for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes $Λ$.
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