<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>36</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Xiyue Han</style></author><author><style face="normal" font="default" size="100%">Alexander Schied</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Robust Faber--Schauder approximation based on discrete observations of an antiderivative</style></title><secondary-title><style face="normal" font="default" size="100%">Mathematics of Operations Research</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">Accepted</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">https://arxiv.org/pdf/2211.11907.pdf</style></url></web-urls></urls><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">We study the problem of reconstructing the Faber--Schauder coefficients of a continuous function&amp;nbsp;f&amp;nbsp;from discrete observations of its antiderivative&amp;nbsp;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.</style></abstract></record></records></xml>