We study the problem of reconstructing the Faber--Schauder coefficients of a continuous function $f$ from discrete observations of its antiderivative $F$. Our approach starts with formulating this problem through piecewise quadratic spline interpolation. We then provide a closed-form solution and an in-depth error analysis. These results lead to some surprising observations, which also throw new light on the classical topic of quadratic spline interpolation itself: They show that the well-known instabilities of this method can be located exclusively within the final generation of estimated Faber--Schauder coefficients, which suffer from non-locality and strong dependence on the initial value and the given data. By contrast, all other Faber--Schauder coefficients depend only locally on the data, are independent of the initial value, and admit uniform error bounds. We thus conclude that a robust and well-behaved estimator for our problem can be obtained by simply dropping the final-generation coefficients from the estimated Faber--Schauder coefficients.

翻译：我们研究了基于连续函数$f$的反导数$F$的离散观测来重构其Faber--Schauder系数的问题。我们的方法首先通过分段二次样条插值对此问题进行形式化描述。随后，我们给出了闭式解并进行了深入的误差分析。这些结果带来了一些令人惊讶的发现，同时也为二次样条插值这一经典课题提供了新视角：它们表明，该方法众所周知的非稳定性问题可以完全定位在估计的Faber--Schauder系数的最后一代中，这些系数受非局部性影响，且严重依赖于初值和给定数据。相比之下，所有其他Faber--Schauder系数仅局部依赖于数据，与初值无关，并具有一致的误差界。因此，我们得出结论：通过简单地从估计的Faber--Schauder系数中剔除最后一代系数，即可获得针对该问题的鲁棒且性质良好的估计量。