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Nonuniform sampling and reconstruction in shift-invariant spaces. (English) Zbl 0995.42022

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 \[ f(x)= \sum_{k\in Z}f(k)\text{sinc}(x-k). \] 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 \[ V^p(\phi)=\{ \sum c_k\phi(\cdot -k): \;\{c_k\}\in \ell^p\}, \] 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.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
41A30 Approximation by other special function classes
47A15 Invariant subspaces of linear operators
46N99 Miscellaneous applications of functional analysis
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
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