Quadrature formulas (QFs) based on radial basis functions (RBFs) have become an essential tool for multivariate numerical integration of scattered data. Although numerous works have been published on RBF-QFs, their stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more mature stability theory for global and function-independent RBF-QFs. In particular, we prove stability of these for compactly supported RBFs under certain conditions on the shape parameter and the data points. As an alternative to changing the shape parameter, we demonstrate how the least-squares approach can be used to construct stable RBF-QFs by allowing the number of data points used for numerical integration to be larger than the number of centers used to generate the RBF approximation space. Moreover, it is shown that asymptotic stability of many global RBF-QFs is independent of polynomial terms, which are often included in RBF approximations. While our findings provide some novel conditions for stability of global RBF-QFs, the present work also demonstrates that there are still many gaps to fill in future investigations.

翻译：基于径向基函数（RBF）的求积公式已成为散乱数据多元数值积分的重要工具。尽管已有大量关于RBF求积公式的研究发表，但其稳定性理论仍显不足。本文致力于为全局且函数无关的RBF求积公式建立更完善的稳定性理论奠定基础。具体而言，我们在形状参数和离散数据点满足特定条件下，证明了紧支径向基函数求积公式的稳定性。作为改变形状参数的替代方案，我们展示了如何通过最小二乘法来构造稳定的RBF求积公式，即允许用于数值积分的数据点数量大于用于生成RBF逼近空间的中心数量。此外，研究表明，许多全局RBF求积公式的渐近稳定性与多项式项无关——后者常被纳入RBF逼近中。尽管我们的研究为全局RBF求积公式的稳定性提供了若干新条件，但本文也表明，未来仍有大量空白亟待填补。