Domain-uniform integration lower bounds and local information

Abstract

We study deterministic numerical integration on open subsets of $\mathbb R^d$ of volume one. Let $e_n$ be the minimal worst-case error using at most $n$ function values. For twice continuously differentiable functions with the normalized Lipschitz bounds proposed by Novak, we prove $e_n\geq v_d^{-2/d}n^{-2/d}/(4(d+2))\geq(8π\mathrm e)^{-1}n^{-2/d}$, where $v_d$ is the volume of the Euclidean unit ball. This resolves Novak's conjecture: the lower-bound constant is independent of the dimension and the shape of the domain. A midpoint rule on the cube has the same rate, up to an absolute constant. The proof uses smooth functions that vanish near the sample points and are large elsewhere. The bound therefore remains valid when each sample supplies all local information about the function. For scalar linear targets continuous in the uniform norm, we prove a second result on the global smooth class with prescribed function, gradient, and Hessian bounds: adaptive local information has the same minimal error as values and gradients at points chosen in advance. We give extensions to function recovery, finite integration measures, and anisotropic derivative bounds.