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Exact iterative reconstruction algorithm
for multivariate irregularly sampled functions
in spline-like spaces: The $L^p$-Theory


Authors: Akram Aldroubi and Hans Feichtinger
Journal: Proc. Amer. Math. Soc. 126 (1998), 2677-2686
MSC (1991): Primary 42C15, 46A35, 46E15, 46N99, 47B37
DOI: https://doi.org/10.1090/S0002-9939-98-04319-6
MathSciNet review: 1451788
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Abstract: We prove that the exact reconstruction of a function $s$ from its samples $s(x_i)$ on any ``sufficiently dense" sampling set $\{x_i\}_{i\in \Lambda}$ can be obtained, as long as $s$ is known to belong to a large class of spline-like spaces in $L^p(\cal R^n)$. Moreover, the reconstruction can be implemented using fast algorithms. Since a limiting case is the space of bandlimited functions, our result generalizes the classical Shannon-Whittaker sampling theorem on regular sampling and the Paley-Wiener theorem on non-uniform sampling.


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Additional Information

Akram Aldroubi
Affiliation: National Institutes of Health, Biomedical Engineering and Instrumentation Program, Bethesda, Maryland 20892
Address at time of publication: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: aldroubi@helix.nih.gov, aldroubi@math.vanderbilt.edu

Hans Feichtinger
Affiliation: University of Vienna, Department of Mathematics, Strudlhofg. 4, A-1090 Wien, Austria
Email: fei@tyche.mat.univie.ac.at

DOI: https://doi.org/10.1090/S0002-9939-98-04319-6
Keywords: Non-uniform sampling, shift-invariant spaces, Riesz basis
Received by editor(s): January 28, 1997
Additional Notes: This research was partially supported through the FWF-project S-7001-MAT of the Austrian Science Foundation.
Dedicated: Dedicated to the memory of Richard J. Duffin
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

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