Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces

The problem of reconstructing a function f from a set of non-uniformly distributed, weighted-average sampled values {integral(R)d f(x)psi(xj) (x) dx: j is an element of J} is studied in the context of shift-invariant subspaces of L-2(R-d). Necessary density conditions on the sampling set X = {x(j): j is an element of j) for stable reconstruction are obtained, and fast iterative algorithms are described. The performance of the algorithms are analyzed when the data are corrupted by noise. Estimates are derived for the convergence rates of the algorithms in terms of the sampling density and the diameters of the sampling functionals {Psix(j): X-j is an element of X}. The results provide a mathematical framework for situations arising frequently in applications, e.g., when the sample values are not precise because they are gathered by real-world acquisition devices. (C) 2002 Published by Elsevier Science (USA).