The goal of this paper is threefold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work in proving lower bounds for recently developed new type of discrepancy—the smooth discrepancy. Third, we consider numerical integration in classes, for which we do not impose any smoothness assumptions. We illustrate how nonlinear approximation, in particular greedy approximation, allows us to guarantee some rate of decay of errors of numerical integration even in such a general setting with no smoothness assumptions.

中文翻译：

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平滑假设和无平滑假设下的数值积分和差异

本文的目标是三重的。首先，我们提出了一种从近似理论和差异理论来制定数值积分问题的统一方法。其次，我们展示了在近似理论中开发的技术如何为最近开发的新型差异（平滑差异）证明下界。第三，我们考虑类中的数值积分，对此我们不强加任何平滑假设。我们说明了非线性逼近，特别是贪婪逼近，即使在没有平滑假设的一般设置中，我们也能保证数值积分误差的衰减率。