We study the integration problem on Hilbert spaces of (multivariate) periodic functions. The standard technique to prove lower bounds for the error of quadrature rules uses bump functions and the pigeon hole principle. Recently, several new lower bounds have been obtained using a different technique which exploits the Hilbert space structure and a variant of the Schur product theorem. The purpose of this paper is to (a) survey the new proof technique, (b) show that it is indeed superior to the bump-function technique, and (c) sharpen and extend the results from the previous papers.

翻译：我们研究（多变量）周期函数的希尔伯特空间上的积分问题。证明求积规则误差下界的标准技术使用凸起函数和鸽巢原理。最近，利用一种基于希尔伯特空间结构及舒尔积定理变体的不同技术，获得了若干新的下界。本文旨在：（a）综述这一新的证明技术，（b）证明其确实优于凸起函数技术，以及（c）改进并扩展先前文献中的结果。