In many applications, it is impractical—if not even impossible—to obtain data to fit a known cubature formula (CF). Instead, experimental data is often acquired at equidistant or even scattered locations. In this work, stable (in the sense of nonnegative only cubature weights) high-order CFs are developed for this purpose. These are based on the approach to allow the number of data points N to be larger than the number of basis functions K which are integrated exactly by the CF. This yields an $(N-K)$-dimensional affine linear subspace from which cubature weights are selected that minimize certain norms corresponding to stability of the CF. In the process, two novel classes of stable high-order CFs are proposed and carefully investigated.

在许多应用中，获取数据来拟合已知的体积公式 (CF) 是不切实际的——如果不是不可能的话。相反，实验数据通常是在等距甚至分散的位置获取的。在这项工作中，为此目的开发了稳定的（仅在非负的体积权重的意义上）高阶 CF。这些是基于允许数据点N的数量大于由 CF 精确积分的基函数K的数量的方法。这产生了$(N-钾)$维仿射线性子空间，从中选择体积权重，以最小化与 CF 稳定性相对应的特定范数。在此过程中，提出并仔细研究了两类新的稳定高阶 CF。