Frames and Numerical Approximation II: Generalized Sampling
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Frames and Numerical Approximation II: Generalized Sampling Ben Adcock1 · Daan Huybrechs2 Received: 19 February 2019 / Revised: 10 July 2020 / Accepted: 15 October 2020 / Published online: 20 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 , 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 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 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.
1 Introduction The approximation of functions in a Hilbert space typically assumes a basis for that space. The non-redundancy of a basis ensures that linear systems associated with approximation problems are non-singular. In addition, suitable structure—ideally the Communicated by Akram Aldroubi.
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Daan Huybrechs [email protected]
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Department of Mathematics, Simon Fraser University, Burnaby, Canada
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Department of Computer Science, KU Leuven, Leuven, Belgium
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Journal of Fourier Analysis and Applications (2020) 26:87
basis is orthonormal, more generally it may be a Riesz basis—renders these systems well-conditioned. There is a unique solution, it is stably computable, and there is a close correspondence between properties of the continuous function and of the coefficients in the representation, for example the Parseval identity. Instead, for a redundant set of functions the corresponding linear systems may be illconditioned or even singular, and uniqueness may be lost. Still, good approximations may exist in the span of the set. It may even be much easier to ensure that this is the case than it is for a basis, and in fact this is a popular approach in a wide range of applications. For example, a basis can be ‘enriched’ by adding a few functions that capture a singularity [10]. A periodic Fourier basis can be augm
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