Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces
In classical sampling theorems it is shown that a function can be recovered from its samples only if it is band limited. For irregular sampling, one could assume that the function belongs to some shift invariant space \(V^p(\phi)=\{f(\cdot)=\sum_{k\in{\mathbb Z}} c_k\phi(\cdot -k): (c_k)\in\ell^p({\mathbb Z})\}\) where \(\phi\) is some generating function (like in a multiresolution analysis). For example, if \(\phi\) is the indicator function \(\chi[0,1]\), then \(V^p(\phi)\) is the space of piecewise constant functions in \(L^p({\mathbb R})\). It is sufficient that there is one sample in each interval to reconstruct the function. The set of sample points \(X\) has to have a density \(D_\phi(X)=\inf\#(X\cap[k,k+1))\geq 1\). Moreover, it is important for practical computations that the reconstruction is stable, meaning that \(\|f\|_p\leq C(\sum_{x_j\in X}|f(x_j)|^p)^{1/p}\). This paper discusses this kind of results for \(\phi\) equal to the spline function \(\beta^n=\chi[0,1]*\chi[0,1]*\cdots*\chi[0,1]\) or when it is compactly supported. A general sufficient condition is given for stable reconstruction from irregular sampling in \(V^p(\beta^n)\) and necessary density conditions are given for general shift invariant subspaces \(V^p(\phi)\) when \(\phi\) is compactly supported.
scientific article; zbMATH DE number 1445628
scientific article