  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.
