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Approximation of Non-decaying Signals from Shift-Invariant Subspaces

Abstract

In our recent work, the sampling and reconstruction of non-decaying signals, modeled as members of weighted-\(L_p\) spaces, were shown to be stable with an appropriate choice of the generating kernel for the shift-invariant reconstruction space. In this paper, we extend the Strang–Fix theory to show that, for d-dimensional signals whose derivatives up to order L are all in some weighted-\(L_p\) space, the weighted norm of the approximation error can be made to go down as \(O(h^L)\) when the sampling step h tends to 0. The sufficient condition for this decay rate is that the generating kernel belongs to a particular hybrid-norm space and satisfies the Strang–Fix conditions of order L. We show that the \(O(h^L)\) behavior of the error is attainable for both approximation schemes using projection (when the signal is prefiltered with the dual kernel) and interpolation (when a prefilter is unavailable). The requirement on the signal for the interpolation method, however, is slightly more stringent than that of the projection because we need to increase the smoothness of the signal by a margin of \(d/p+\varepsilon \), for arbitrary \(\varepsilon >0\). This extra amount of derivatives is used to make sure that the direct sampling is stable.

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Notes

  1. 1.

    To be precise, when f is multivariate, \(f^{(L)}\) is the summation of (the moduli of) all partial derivatives of order L of f.

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Author information

Correspondence to Ha Q. Nguyen.

Additional information

This research was funded by the Swiss National Science Foundation under Grant No. 200020-162343.

Ha Q. Nguyen: The majority of this work was done when he was with the Biomedical Imaging Group, École Polytechnique Fédérale de Lausanne (EPFL), Station 17, 1015, Lausanne, Switzerland.

Communicated by Chris Heil.

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Nguyen, H.Q., Unser, M. Approximation of Non-decaying Signals from Shift-Invariant Subspaces. J Fourier Anal Appl 25, 633–660 (2019) doi:10.1007/s00041-018-9622-6

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Keywords

  • Approximation theory
  • Strang–Fix conditions
  • Shift-invariant spaces
  • Spline interpolation
  • Weighted \(L_p\)
  • Weighted Sobolev spaces
  • Hybrid-norm spaces