Wiener 代数子空间中多元积分的强易处理性
数值分析
2023-06-05 v1 数值分析
摘要
基于作者近期工作，我们证明在 [ 0 , 1 ) d [0,1)^d [ 0 , 1 ) d F d : = { f ∈ C ( [ 0 , 1 ) d ) | ∥ f ∥ : = ∑ k ∈ Z d ∣ f ^ ( k ) ∣ max ( w i d t h ( s u p p ( k ) ) , min j ∈ s u p p ( k ) log ∣ k j ∣ ) < ∞ } , F_d:=\left\{ f\in C([0,1)^d)\:\middle| \: \|f\|:=\sum_{\boldsymbol{k}\in \mathbb{Z}^{d}}|\hat{f}(\boldsymbol{k})|\max\left(\mathrm{width}(\mathrm{supp}(\boldsymbol{k})),\min_{j\in \mathrm{supp}(\boldsymbol{k})}\log |k_j|\right)<\infty \right\}, F d := { f ∈ C ([ 0 , 1 ) d ) ∥ f ∥ := k ∈ Z d ∑ ∣ f ^ ( k ) ∣ max ( width ( supp ( k )) , j ∈ supp ( k ) min log ∣ k j ∣ ) < ∞ } , f ^ ( k ) \hat{f}(\boldsymbol{k}) f ^ ( k ) f f f k \boldsymbol{k} k s u p p ( k ) : = { j ∈ { 1 , … , d } ∣ k j ≠ 0 } \mathrm{supp}(\boldsymbol{k}):=\{j\in \{1,\ldots,d\}\mid k_j\neq 0\} supp ( k ) := { j ∈ { 1 , … , d } ∣ k j = 0 } w i d t h : 2 { 1 , … , d } → { 1 , … , d } \mathrm{width}: 2^{\{1,\ldots,d\}}\to \{1,\ldots,d\} width : 2 { 1 , … , d } → { 1 , … , d } w i d t h ( u ) : = max j ∈ u j − min j ∈ u j + 1 , \mathrm{width}(u):=\max_{j\in u}j-\min_{j\in u}j+1, width ( u ) := j ∈ u max j − j ∈ u min j + 1 , u ⊆ { 1 , … , d } u\subseteq \{1,\ldots,d\} u ⊆ { 1 , … , d } w i d t h ( ∅ ) : = 1 \mathrm{width}(\emptyset):=1 width ( ∅ ) := 1 p p p k ∈ Z d \boldsymbol{k}\in \mathbb{Z}^d k ∈ Z d w i d t h ( s u p p ( k ) ) \mathrm{width}(\mathrm{supp}(\boldsymbol{k})) width ( supp ( k ))
引用
@article{arxiv.2306.01541,
title = {Strong tractability for multivariate integration in a subspace of the Wiener algebra},
author = {Takashi Goda},
journal= {arXiv preprint arXiv:2306.01541},
year = {2023}
}
备注
8 pages