Laguerre- and Laplace-weighted integration of mixed-smoothness functions

Abstract

We investigate the approximation of generalized Laguerre- or Laplace-weighted integrals over ${\mathbb{R}}_{+}^d$ or ${\mathbb{R}}^d$ of functions from generalized Laguerre- or Laplace-weighted Sobolev spaces of mixed smoothness, respectively. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$ integration nodes for functions from these spaces. The upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic corners or hyperbolic crosses in the function domain ${\mathbb{R}}_{+}^d$ or ${\mathbb{R}}^d$, respectively.