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Frames and Numerical Approximation II: Generalized Sampling

Abstract

In a previous paper (Adcock and Huybrechs in SIAM Rev 61(3):443–473, 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 \(\sqrt{\epsilon }\), where \(\epsilon \) 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 f from a wide class of linear operators acting on f, 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 \(\epsilon \) rather than \(\sqrt{\epsilon }\). 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.

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Acknowledgements

A preliminary version of this work was presented during the Research Cluster on “Computational Challenges in Sparse and Redundant Representations” at ICERM in November 2014. The authors would like to thank all the participants for the useful discussions and feedback received during the programme. The first author would also like to thank Juan M. Cardenas and Sebastian Moraga. The first author is supported by NSERC Grant 611675, as well as an Alfred P. Sloan Research Fellowship The second author is supported by FWO-Flanders projects G.0641.11, G.A004.14 and by KU Leuven Project C14/15/055.

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Correspondence to Daan Huybrechs.

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Communicated by Akram Aldroubi.

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Adcock, B., Huybrechs, D. Frames and Numerical Approximation II: Generalized Sampling. J Fourier Anal Appl 26, 87 (2020). https://doi.org/10.1007/s00041-020-09796-w

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  • DOI: https://doi.org/10.1007/s00041-020-09796-w