Strong tractability for multivariate integration in a subspace of the Wiener algebra (2306.01541v1)

Abstract: 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.