Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces
The paper concerns non-uniform sampling, and the setup is as follows. Assume that the continuous function \(\phi: \mathbb{R}^d\to C\) (belonging to the Wiener space) generates a Riesz basis for the shift-invariant space \[ V^2=\Bigl \{ \sum_{k\in Z^d} c_k \phi(\cdot -k): \{c_k\}\in \ell^2\Bigr\}. \] Consider furthermore some points \(\{x_j\}\subset \mathbb{R}^d\) and associated bounded functions \(\psi_{x_j}\) such that supp \(\psi_{x_j}\subset x_j+ [-a,a]^d\) for some \(a>0\), and such that for some constants \(c,C>0\), \[ c||f||^2 \leq \sum_j |\langle f, \psi_{x_j}\rangle |^2 \leq C||f||^2, \;\forall f\in V^2. \] Note that this is different from a frame condition because the functions \(\psi_{x_j}\) are not assumed to belong to \(V^2\). The paper presents an iterative algorithm to recover a function \(f\in V^2\) from the data \(\{\langle f, \psi_{x_j}\rangle \}\), provided that the points \(\{x_j\}\) are sufficiently dense and the support of \(\psi_{x_j}\) is sufficiently small. The convergence is geometric, and the rate of convergence is estimated.
scientific article; zbMATH DE number 1891397
scientific article