Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples (Q2627890)

In this interesting paper, the authors reconstruct a compactly supported function \(f\in L^2({\mathbb R}^d)\) from pointwise measurements of its Fourier transform \(\hat f\). Let \(H:= \{f \in L^2({\mathbb R}^d):\,{\mathrm{supp}}\,f \subseteq E\}\), where \(E\subset \mathbb R^d\) is compact. First the authors improve a result of \textit{K.~Gröchenig} [Math. Comput. 59, No.199, 181--194 (1992; Zbl 0794.46009)] and show the following: If \(\Omega \subset \mathbb R^d\) is a countable sampling set with certain density and if \(E\) is convex and symmetric too, then there exist weights \(\mu_{\omega}> 0\) such that \(\{\sqrt{\mu_{\omega}}\, e_{\omega}:\, \omega \in \Omega\}\) is a weighted Fourier frame of \(H\), where \(e_{\omega} := \exp(2\pi \,i\,\omega \cdot x)\, 1_{E}(x)\). In other words, there exist frame bounds \(A\), \(B>0\) such that for all \(f\in H\) \[ A\, \|f\|^2 \leq \sum_{\omega \in \Omega} \mu_{\omega}\,|{\hat f}(\omega)|^2 \leq B\, \|f \|^2\,. \] Here one can choose the weights \(\mu_{\omega}\) as certain measures of Voronoi regions. With this result, the authors provide a density condition which is both sharp and dimensionless. However, this result does not give rise to explicit frame bounds. Therefore, in a second theorem the authors present explicit frame bounds under a less stringent density condition. Using these results, stable and accurate recovery of \(f \in H\) is shown. Several examples illustrate the effectiveness of this approach.