Multivariate integration in C^\infty([0,1]^d) is not strongly tractable

TL;DR

This paper proves that multivariate integration for infinitely differentiable functions on the unit cube remains intractable, extending known results to the case of infinite smoothness.

Contribution

It provides a partial proof that multivariate integration in $C^ abla([0,1]^d)$ is not strongly tractable, supporting Woźniakowski's conjecture.

Findings

Multivariate integration in $C^ abla([0,1]^d)$ is not strongly tractable.

The result extends intractability from finite to infinite smoothness.

Supports the conjecture that smoothness does not alleviate intractability.

Abstract

It has long been known that the multivariate integration problem for the unit ball in $C^r([0,1]^d)$ is intractable for fixed finite $r$. H. Wo\'zniakowski has recently conjectured that this is true even if $r=\infty$. This paper establishes a partial result in this direction. We prove that the multivariate integration problem, for infinitely differential functions all of whose variables are bounded by one, is not strongly tractable.