Nonuniform sampling and reconstruction in shift-invariant spaces (Q2780621)

The sampling problem concerns how a function can be recovered based on knowledge of some of its function values. For this to be possible we need additional information about the function: a classical example is that every function in \(f\in L^2(R)\) whose Fourier transform is supported in \([-1/2,1/2]\) can be recovered from the samples \(\{f(k)\}_{k\in Z}\) via NEWLINE\[NEWLINEf(x)= \sum_{k\in Z}f(k)\text{sinc}(x-k). NEWLINE\]NEWLINE Non-uniform sampling concerns recovering of functions based on samples \(\{f(x_k)\}\), where \(\{x_k\}\) is a sequence in \(R\). The present paper, which is a combined survey and research paper, addresses the sampling problem in shift-invariant spaces, i.e., spaces of the type NEWLINE\[NEWLINEV^p(\phi)=\{ \sum c_k\phi(\cdot -k): \;\{c_k\}\in \ell^p\},NEWLINE\]NEWLINE and weighted versions. Several steps have to be taken: first, one needs conditions on \(\phi\) such that \(V^p(\phi)\) is well defined, and second, one needs to assure that the sampling problem makes sense (for this, the space has to consist of continuous functions). It turns out that a sufficient condition on \(\phi\) is that it belongs to a certain Wiener amalgam space; this will make \(V^p(\phi)\) a subspace of \(L^p\). Iterative algorithms for reconstruction of functions based on samples are provided.