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We investigate multivariate integration for a space of infinitely timesdifferentiable functions $\mathcal{F}_{s, \boldsymbol{u}} := \{f \in C^\infty[0,1]^s \mid \| f \|_{\mathcal{F}_{s, \boldsymbol{u}}} < \infty \}$, where $\|f \|_{\mathcal{F}_{s, \boldsymbol{u}}} := \sup_{\boldsymbol{\alpha} =(\alpha_1, \dots, \alpha_s) \in \mathbb{N}_0^s}\|f^{(\boldsymbol{\alpha})}\|_{L^1}/\prod_{j=1}^s u_j^{\alpha_j}$,$f^{(\boldsymbol{\alpha})} := \frac{\partial^{|\boldsymbol{\alpha}|}}{\partialx_1^{\alpha_1} \cdots \partial x_s^{\alpha_s}}f$ and $\boldsymbol{u} =\{u_j\}_{j \geq 1}$ is a sequence of positive decreasing weights. Let $e(n,s)$be the minimal worst-case error of all algorithms that use $n$ function valuesin the $s$-variate case. We prove that for any $\boldsymbol{u}$ and $s$considered $e(n,s) \leq C(s) \exp(-c(s)(\log{n})^2)$ holds for all $n$, where$C(s)$ and $c(s)$ are constants which may depend on $s$. Further we show thatif the weights $\boldsymbol{u}$ decay sufficiently fast then there exist some$1 < p < 2$ and absolute constants $C$ and $c$ such that $e(n,s) \leq C\exp(-c(\log{n})^p)$ holds for all $s$ and $n$. These bounds are attained byquasi-Monte Carlo integration using digital nets. These convergence andtractability results come from those for the Walsh space into which$\mathcal{F}_{s, \boldsymbol{u}}$ is embedded.

Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions