Frames and numerical approximation. II: Generalized sampling
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Publication:2226990
DOI10.1007/S00041-020-09796-WzbMATH Open1462.42048arXiv1802.01950OpenAlexW3105082494MaRDI QIDQ2226990
Author name not available (Why is that?)
Publication date: 9 February 2021
Published in: (Search for Journal in Brave)
Abstract: In a previous paper [Adcock & Huybrechs, 2019] we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormous flexibility compared to using a basis, but ill-conditioning often prevents the numerical computation of best approximations. We showed that, in spite of said ill-conditioning, approximations with regularization may still provide accuracy up to order , where is a small truncation threshold. When using frames, i.e. complete systems that are generally redundant but which provide infinite representations with coefficients of bounded norm, this accuracy can actually be achieved for all functions in a space. Here, we generalize that setting in two ways. We assume information or samples from from a wide class of linear operators acting on , rather than inner products associated with the best approximation projection. This enables the analysis of fully discrete approximations based, for instance, on function values only. Next, we allow oversampling, leading to least-squares approximations. We show that this leads to much improved accuracy on the order of rather than . Overall, we demonstrate that numerical function approximation using redundant representations may lead to highly accurate approximations in spite of having to solve ill-conditioned systems of equations.
Full work available at URL: https://arxiv.org/abs/1802.01950
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