Multi- and Infinite-variate Integration and $L^2$-Approximation on Hilbert Spaces with Gaussian Kernels
Michael Gnewuch, Klaus Ritter, Robin Rüßmann
TL;DR
The paper develops a rigorous framework for worst-case integration and $L^2$-approximation on RKHSs with tensor-product Gaussian kernels, covering both finite and infinite variates. A central contribution is a constructive isometric transference to Hermite spaces, enabling explicit algorithmic correspondences and tight bounds based on well-understood Hermite-space results. It establishes polynomial decay rates for infinite-variable problems and characterizes exponential convergence and weak tractability in the finite-variable regime, including precise conditions on the parameter sequences governing the kernels. The work provides practical guidelines for designing and analyzing high-dimensional quadrature and approximation schemes, with implications for complex stochastic models and infinite-dimensional function spaces.
Abstract
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.