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In this paper we analyze the approximation of multivariate integrals over theEuclidean plane for functions which are analytic. We show explicit upper boundswhich attain the exponential rate of convergence. We use an infinite grid withdifferent mesh sizes and lengths in each direction to sample the function, andthen truncate it. In our analysis, the mesh sizes and the truncated domain arechosen by optimally balancing the truncation error and the discretizationerror.  This paper derives results in comparable function space settings, extended to$\R^s$, as which were recently obtained in the unit cube by Dick, Larcher,Pillichshammer and Wo{\'z}niakowski (2011).  They showed that both lattice rules and regular grids, with different meshsizes in each direction, attain exponential rates, hence motivating us toanalyze only cubature formula based on regular meshes.  We further also amend the analysis of older publications, e.g., Sloan andOsborn (1987) and Sugihara (1987), using lattice rules on $\R^s$ by taking thetruncation error into account and extending them to take the anisotropy of thefunction space into account.

Multivariate integration over $\R^s$ with exponential rate of convergence