Building upon recent work by the author, we prove that multivariate integration in the following subspace of the Wiener algebra over $[0,1)^d$ is strongly polynomially tractable: \[ F_d:=\left\{ f\in C([0,1)^d)\:\middle| \: \|f\|:=\sum_{\boldsymbol{k}\in \mathbb{Z}^{d}}|\hat{f}(\boldsymbol{k})|\max\left(\mathrm{width}(\mathrm{supp}(\boldsymbol{k})),\min_{j\in \mathrm{supp}(\boldsymbol{k})}\log |k_j|\right)<\infty \right\},\] with $\hat{f}(\boldsymbol{k})$ being the $\boldsymbol{k}$-th Fourier coefficient of $f$, $\mathrm{supp}(\boldsymbol{k}):=\{j\in \{1,\ldots,d\}\mid k_j\neq 0\}$, and $\mathrm{width}: 2^{\{1,\ldots,d\}}\to \{1,\ldots,d\}$ being defined by \[ \mathrm{width}(u):=\max_{j\in u}j-\min_{j\in u}j+1,\] for non-empty subset $u\subseteq \{1,\ldots,d\}$ and $\mathrm{width}(\emptyset):=1$. Strong polynomial tractability is achieved by an explicit quasi-Monte Carlo rule using a multiset union of Korobov's $p$-sets. We also show that, if we replace $\mathrm{width}(\mathrm{supp}(\boldsymbol{k}))$ with 1 for all $\boldsymbol{k}\in \mathbb{Z}^d$ in the above definition of norm, multivariate integration is polynomially tractable but not strongly polynomially tractable.

翻译：基于作者近期的工作，我们证明在 $[0,1)^d$ 上Wiener代数的如下子空间中的多元积分是强多项式易处理的：\[ F_d:=\left\{ f\in C([0,1)^d)\:\middle| \: \|f\|:=\sum_{\boldsymbol{k}\in \mathbb{Z}^{d}}|\hat{f}(\boldsymbol{k})|\max\left(\mathrm{width}(\mathrm{supp}(\boldsymbol{k})),\min_{j\in \mathrm{supp}(\boldsymbol{k})}\log |k_j|\right)<\infty \right\},\] 其中 $\hat{f}(\boldsymbol{k})$ 是 $f$ 的第 $\boldsymbol{k}$ 个傅里叶系数，$\mathrm{supp}(\boldsymbol{k}):=\{j\in \{1,\ldots,d\}\mid k_j\neq 0\}$，而 $\mathrm{width}: 2^{\{1,\ldots,d\}}\to \{1,\ldots,d\}$ 定义为 \[ \mathrm{width}(u):=\max_{j\in u}j-\min_{j\in u}j+1,\] 对于非空子集 $u\subseteq \{1,\ldots,d\}$，且 $\mathrm{width}(\emptyset):=1$。该强多项式易处理性通过使用Korobov $p$-集的多集并的显式拟蒙特卡罗规则实现。我们还证明，若在上述范数定义中将所有 $\boldsymbol{k}\in \mathbb{Z}^d$ 的 $\mathrm{width}(\mathrm{supp}(\boldsymbol{k}))$ 替换为1，则多元积分是多项式易处理的，但非强多项式易处理的。