Weighted Integral of Infinitely Differentiable Multivariate Functions is Exponentially Convergent

Abstract

We study the problem of a weighted integral of infinitely differentiable multivariate functions defined on the unit cube with the $L$_∞-norm of partial derivative of all orders bounded by 1. We consider the algorithms that use finitely many function values as information (called standard information). On the one hand, we obtained that the interpolatory quadratures based on the extended Chebyshev nodes of the second kind have almost the same quadrature weights. On the other hand, by using the  Smolyak algorithm with the above interpolatory quadratures, we proved that the weighted integral problem is of exponential convergence in the worst case setting.