Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the approximation or integration of functions defined on $D_d \subset R^d$ and only assume that $D_d$ is a bounded Lipschitz domain. Some results are even more general. We study three different concepts to measure the complexity: order of convergence, asymptotic constant, and explicit uniform bounds, i.e., bounds that hold for all $n$ (number of pieces of information) and all (normalized) domains. It is known for many problems that the order of convergence of optimal algorithms does not depend on the domain $D_d \subset R^d$. We present examples for which the following statements are true: 1) Also the asymptotic constant does not depend on the shape of $D_d$ or the imposed boundary values, it only depends on the volume of the domain. 2) There are explicit and uniform lower (or upper, respectively) bounds for the error that are only slightly smaller (or larger, respectively) than the asymptotic error bound.

中文翻译：

---

一般域上函数的算法和复杂性

数值分析中的误差界限和复杂性界限以及基于信息的复杂性通常适用于在非常简单的域上定义的函数，例如立方体、环面或球体。我们研究了定义在 $D_d \subset R^d$ 上的函数的近似或积分的最佳误差界限，并且仅假设 $D_d$ 是有界 Lipschitz 域。有些结果甚至更普遍。我们研究了三个不同的概念来衡量复杂性：收敛阶数、渐近常数和显式统一边界，即所有 $n$（信息的数量）和所有（归一化）域的边界。众所周知，最优算法的收敛阶次不依赖于域$D_d \subset R^d$。我们提供了以下陈述正确的例子：1) 此外，渐近常数不依赖于 $D_d$ 的形状或强加的边界值，它只取决于域的体积。2) 有明确和统一的误差下限（或上限），仅比渐近误差界略小（或大）。