Biomedical Imaging GroupSTI
English only   BIG > Publications > Sampling Theory

 Home Page
 News & Events
 Tutorials and Reviews
 Download Algorithms

 All BibTeX References

A Sampling Theory for Non-Decaying Signals

H.Q. Nguyen, M. Unser

Applied and Computational Harmonic Analysis, vol. 43, no. 1, pp. 76-93, July 2017.

The classical assumption in sampling and spline theories is that the input signal is square-integrable, which prevents us from applying such techniques to signals that do not decay or even grow at infinity. In this paper, we develop a sampling theory for multidimensional non-decaying signals living in weighted Lp spaces. The sampling and reconstruction of an analog signal can be done by a projection onto a shift-invariant subspace generated by an interpolating kernel. We show that, if this kernel and its biorthogonal counterpart are elements of appropriate hybrid-norm spaces, then both the sampling and the reconstruction are stable. This is an extension of earlier results by Aldroubi and Gröchenig. The extension is required because it allows us to develop the theory for the ideal sampling of non-decaying signals in weighted Sobolev spaces. When the d-dimensional signal and its dp + ε derivatives, for arbitrarily small ε > 0, grow no faster than a polynomial in the Lp sense, the sampling operator is shown to be bounded even without a sampling kernel. As a consequence, the signal can also be interpolated from its samples with a nicely behaved interpolating kernel.

AUTHOR="Nguyen, H.Q. and Unser, M.",
TITLE="A Sampling Theory for Non-Decaying Signals",
JOURNAL="Applied and Computational Harmonic Analysis",

© 2017 Elsevier. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from Elsevier.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.