Strong tractability for multivariate integration in a subspace of the Wiener algebra

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 $\hat{f}(\mathbf{k})$ being the $\mathbf{k}$-th Fourier coefficient of $f$, $\mathrm{supp}(\mathbf{k}):=\{j\in \{1,\dots ,d\}\mid k_j\ne 0\}$, and $\mathrm{width}:2^{\{1,\dots ,d\}}\to \{1,\dots ,d\}$ being defined by \[ \mathrm{width}(u):=\max_{j\in u}j-\min_{j\in u}j+1,\] for non-empty subset $u\subseteq \{1,\dots ,d\}$ and $\mathrm{width}(\varnothing ):=1$. 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 $\mathrm{width}(\mathrm{supp}(\mathbf{k}))$ with 1 for all $\mathbf{k}\in {\mathbb{Z}}^d$ in the above definition of norm, multivariate integration is polynomially tractable but not strongly polynomially tractable.

8 pages