  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.

Numerical Analysis;Numerical Analysis;Statistical Finance