论文标题
多项式障碍可在具有绝对收敛傅里叶序列的未加权功能空间中集成
Polynomial tractability for integration in an unweighted function space with absolutely convergent Fourier series
论文作者
论文摘要
在本注中，我们证明了以下具有绝对收敛的傅里叶系列\ [f_d：= \ left \ {f \ in l^2（[0,1）^d）\：\ middle |的功能空间。 \：\ | f \ |：= \ sum _ {\ boldsymbol {k} \ in \ mathbb {z}^d} | \ hat {f}（\ boldsymbol {k}）| \ max \ left（1，\ min_ {j \ in \ mathrm {spep}（\ boldsymbol {k}）} \ log | k_j | \ right）<\ infty \ right \} \]，$ \ hat {f}（f}（f}）（\ boldsymbol { $ f $和$ \ mathrm {supp}（\ boldsymbol {k}）：= \ {j \ in \ {1，\ ldots，d \} \ mid k_j \ neq 0 \} $是多项式拖动，用于在erast-case设置中进行多变量集成。此处多项式障碍性意味着使最差的误差小于或等于公差$ \ varepsilon $仅相对于$ \ varepsilon^{ - 1} $和$ d $多样化，所需的最小功能评估数量。重要的是要指出，功能空间$ f_d $未加权，也就是说，所有变量都对函数规范均等贡献。我们的障碍结果与文献中研究的大多数未加权整合问题的结果相反，在文献中，多项式障碍性不存在，并且该问题受到维数的诅咒。我们的证明是建设性的，从某种意义上说，我们提供了一个明确的准蒙特·卡洛规则，该规则达到了所需的最坏情况误差。
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.
