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：我们研究关于（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了碰撞函数和鸽巢原理。近年来，利用一种不同的技术获得了几个新的下界，该技术利用了希尔伯特空间结构和Schur乘积定理的一个变体。本文的目的是：(a) 综述这种新的证明技术，(b) 展示它确实优于碰撞函数技术，以及(c) 深化并扩展先前论文的结果。