Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The 𝐿^{𝑝}-theory

Aldroubi, Akram; Feichtinger, Hans

doi:10.1090/S0002-9939-98-04319-6

Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The $L^p$-theory
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by Akram Aldroubi and Hans Feichtinger PDF

Proc. Amer. Math. Soc. 126 (1998), 2677-2686 Request permission

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 $L^p (\mathcal {R}^n)$. 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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