Relevant sampling in finitely generated shift-invariant spaces

Abstract

We consider random sampling in finitely generated shift-invariant spaces $V(\Phi )\subset {\mathrm{L}}^2({\mathbb{R}}^n)$ generated by a vector $\Phi =({\phi }_1,\dots ,{\phi }_r)\in {\mathrm{L}}^2({\mathbb{R}}^n{)}^r$. Following the approach introduced by Bass and Gröchenig, we consider certain relatively compact subsets $V_{R,\delta }(\Phi )$ of such a space, defined in terms of a concentration inequality with respect to a cube with side lengths $R$. Under very mild assumptions on the generators, we show that for $R$ sufficiently large, taking $O(R^{n}log(R^{n^2/{\alpha }^{\prime }}))$ many random samples (taken independently uniformly distributed within $C_R$) yields a sampling set for $V_{R,\delta }(\Phi )$ with high probability. Here ${\alpha }^{\prime }\le n$ is a suitable constant.We give explicit estimates of all involved constants in terms of the generators ${\phi }_1,\dots ,{\phi }_r$.

17 pages

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