We study lower bounds on the worst-case error of numerical integration in tensor product spaces. As reference we use the $N$-th minimal error of linear rules that use $N$ function values. The information complexity is the minimal number $N$ of function evaluations that is necessary such that the $N$-th minimal error is less than a factor $\varepsilon$ times the initial error. We are interested to which extent the information complexity depends on the number $d$ of variables of the integrands. If the information complexity grows exponentially fast in $d$, then the integration problem is said to suffer from the curse of dimensionality. Under the assumption of the existence of a worst-case function for the uni-variate problem we present two methods for providing good lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function. This method can be seen as a generalization of the method of decomposable reproducing kernels, that is often successfully applied when integration in Hilbert spaces with a reproducing kernel is studied. The second method, although only applicable for positive quadrature rules, has the advantage, that it does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. The methods presented can be applied to problems beyond the Hilbert space setting. For demonstration purposes we apply them to several examples, notably to uniform integration over the unit-cube, weighted integration over the whole space, and integration of infinitely smooth functions over the cube. Some of these results have interesting consequences in discrepancy theory.

翻译：我们研究了张量积空间中数值积分最坏情况误差的下界。我们以使用$N$个函数值的线性规则的$N$阶最小误差作为参考。信息复杂度是指使得$N$阶最小误差小于初始误差的$\varepsilon$倍所需的最小函数求值次数$N$。我们关注信息复杂度在多大程度上依赖于被积函数的变量个数$d$。如果信息复杂度随$d$呈指数增长，则称该积分问题存在维数灾难。在假设单变量问题存在最坏情况函数的前提下，我们提出了两种给出信息复杂度良好下界的方法。第一种方法基于最坏情况函数的适当分解，可视为可分解再生核方法的推广——该方法在带再生核的希尔伯特空间积分问题研究中常被成功应用。第二种方法虽仅适用于正求积规则，但其优势在于无需对最坏情况函数进行适当分解，而是通过对最坏情况函数进行样条逼近，且适用于解析函数族。所提出的方法可应用于希尔伯特空间框架之外的问题。为演示目的，我们将这些方法应用于多个实例，特别包括单位立方体上的均匀积分、整个空间上的加权积分以及立方体上无穷光滑函数的积分。其中部分结果在偏差理论中具有有趣启示。