标题： 多变量和无限变量积分以及在具有高斯核的希尔伯特空间上的$L^2$-逼近 显示英文标题

标题： Multi- and Infinite-variate Integration and $L^2$-Approximation on Hilbert Spaces with Gaussian Kernels

摘要： 我们研究在最坏情况下，基于函数值的确定性线性算法的积分和$L^2$-逼近。 基础函数空间是一个具有张量积形式高斯核的再生核希尔伯特空间。 在无限变量情况下，对于这两个计算问题，我们建立了$n$-最小误差多项式收敛速率的匹配上下界。 在多变量情况下，我们改进了积分问题的若干可解性结果。 对于证明，我们建立了以下传递结果以及显式构造：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。 显示英文摘要

摘要： 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.