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Frame approximation with bounded coefficients

Abstract

Due to their flexibility, frames of Hilbert spaces are attractive alternatives to bases in approximation schemes for problems where identifying a basis is not straightforward or even feasible. Computing a best approximation using frames, however, can be challenging since it requires solving an ill-conditioned linear system. One consequence of this ill-conditioning is that the coefficients of such a frame approximation can grow large. In this paper, we resolve this issue by introducing two methods for frame approximation that possess bounded coefficients. As we show, these methods typically lead to little or no deterioration in the approximation accuracy, but successfully avoid the large coefficients inherent to previous approaches, thus making them attractive in situations where large coefficients are undesirable. We also present theoretical analysis to support these conclusions.

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Acknowledgements

The question of frame approximation with bounded coefficients was first raised during a talk by the first author at the Oberwolfach conference on ‘Multiscale and High-Dimensional Problems’. The authors would like to thank Ingrid Daubechies for raising this question. They would also like to thank Daan Huybrechs for helpful comments and suggestions.

Funding

This work was supported by NSERC through grant 611675, as well as through the PIMS CRG on ‘High-dimensional Data Analysis’.

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Correspondence to Ben Adcock.

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Communicated by: Holger Rauhut

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Adcock, B., Seifi, M. Frame approximation with bounded coefficients. Adv Comput Math 47, 4 (2021). https://doi.org/10.1007/s10444-020-09820-z

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  • DOI: https://doi.org/10.1007/s10444-020-09820-z

Keywords

  • Frames
  • Function approximation
  • Ill-conditioning
  • Singular value decomposition

Mathematics Subject Classification (2010)

  • 42C15
  • 42C30
  • 41A10
  • 65T40