In this note, we prove that the following function space with absolutely convergent Fourier series \[ F_d:=\left\{ f\in L^2([0,1)^d)\:\middle| \: \|f\|:=\sum_{\boldsymbol{k}\in \mathbb{Z}^d}|\hat{f}(\boldsymbol{k})| \max\left(1,\min_{j\in \mathrm{supp}(\boldsymbol{k})}\log |k_j|\right) <\infty \right\}\] with $\hat{f}(\boldsymbol{k})$ being the $\boldsymbol{k}$-th Fourier coefficient of $f$ and $\mathrm{supp}(\boldsymbol{k}):=\{j\in \{1,\ldots,d\}\mid k_j\neq 0\}$ is polynomially tractable for multivariate integration in the worst-case setting. Here polynomial tractability means that the minimum number of function evaluations required to make the worst-case error less than or equal to a tolerance $\varepsilon$ grows only polynomially with respect to $\varepsilon^{-1}$ and $d$. It is important to remark that the function space $F_d$ is unweighted, that is, all variables contribute equally to the norm of functions. Our tractability result is in contrast to those for most of the unweighted integration problems studied in the literature, in which polynomial tractability does not hold and the problem suffers from the curse of dimensionality. Our proof is constructive in the sense that we provide an explicit quasi-Monte Carlo rule that attains a desired worst-case error bound.

翻译：本文证明，以下具有绝对收敛傅里叶级数的函数空间 \[ F_d:=\left\{ f\in L^2([0,1)^d)\:\middle| \: \|f\|:=\sum_{\boldsymbol{k}\in \mathbb{Z}^d}|\hat{f}(\boldsymbol{k})| \max\left(1,\min_{j\in \mathrm{supp}(\boldsymbol{k})}\log |k_j|\right) <\infty \right\}\] 其中 $\hat{f}(\boldsymbol{k})$ 是 $f$ 的第 $\boldsymbol{k}$ 个傅里叶系数，$\mathrm{supp}(\boldsymbol{k}):=\{j\in \{1,\ldots,d\}\mid k_j\neq 0\}$，在最坏情形设定下对多元积分是多项式可处理的。此处多项式可处理性意味着，使得最坏情形误差不超过容差 $\varepsilon$ 所需的最小函数求值次数相对于 $\varepsilon^{-1}$ 和 $d$ 仅以多项式增长。值得强调的是，函数空间 $F_d$ 是无加权的，即所有变量对函数范数的贡献均等。本文的可处理性结果与文献中研究的多数无加权积分问题形成对比，后者不具备多项式可处理性且面临维度诅咒问题。本文的证明是构造性的，我们提供了显式的拟蒙特卡洛规则，该规则可实现所需的最坏情形误差界。