We study integration and $L^2$-approximation in the worst-case setting for deterministic linear algorithms based on function evaluations. The underlying function space is a reproducing kernel Hilbert space with a Gaussian kernel of tensor product form. In the infinite-variate case, for both computational problems, we establish matching upper and lower bounds for the polynomial convergence rate of the $n$-th minimal error. In the multivariate case, we improve several tractability results for the integration problem. For the proofs, we establish the following transference result together with an explicit construction: Each of the computational problems on a space with a Gaussian kernel is equivalent on the level of algorithms to the same problem on a Hermite space with suitable parameters.

翻译：我们研究基于函数值求值的确定性线性算法在最坏情况设定下的积分与$L^2$逼近问题。所涉及的函数空间是具有张量积形式高斯核的再生核希尔伯特空间。在无限变量情形下，针对这两个计算问题，我们为第$n$最小误差的多项式收敛速率建立了匹配的上下界。在多变量情形下，我们改进了积分问题的若干易处理性结果。在证明中，我们通过一个显式构造建立了如下传递性结果：具有高斯核的空间上的每个计算问题，在算法层面上等价于具有适当参数的埃尔米特空间上的同一问题。