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：本文研究从连续函数$f$的反导数$F$的离散观测中重构其Faber-Schauder系数的问题。我们首先通过分段二次样条插值对该问题进行形式化，随后给出闭式解并进行深入误差分析。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：研究表明，该方法众所周知的数值不稳定性完全集中于所估计的Faber-Schauder系数的最后一代，而这代系数存在非局部性、对初始值与给定数据的强依赖性。相比之下，其他所有Faber-Schauder系数仅局部依赖于数据、与初始值无关，并具有一致误差界。由此我们得出结论：仅需从估计的Faber-Schauder系数中剔除最后一代系数，即可获得针对该问题的稳健且性质优良的估计量。