Integration and approximation in cosine spaces of smooth functions
For researchers in high-dimensional numerical analysis, this work provides theoretical guarantees for exponential convergence and tractability in a specific function space, but it is incremental as it extends known techniques to a new setting.
The paper studies multivariate integration and approximation in a weighted reproducing kernel Hilbert space based on half-period cosine functions, proving exponential convergence of worst-case errors and providing necessary and sufficient conditions for tractability.
We study multivariate integration and approximation for functions belonging to a weighted reproducing kernel Hilbert space based on half-period cosine functions in the worst-case setting. The weights in the norm of the function space depend on two sequences of real numbers and decay exponentially. As a consequence the functions are infinitely often differentiable, and therefore it is natural to expect exponential convergence of the worst-case error. We give conditions on the weight sequences under which we have exponential convergence for the integration as well as the approximation problem. Furthermore, we investigate the dependence of the errors on the dimension by considering various notions of tractability. We prove sufficient and necessary conditions to achieve these tractability notions.