Stable nonuniform sampling with weighted Fourier frames and recovery in arbitrary spaces

We present recently devised approach for recovery of compactly supported multivariate functions from nonuniform samples of their Fourier transforms. This is the so-called nonuniform generalized sampling (NUGS), based on a generalized sampling framework which permits an arbitrary choice of the reconstruction space and where nonuniform sampling is modeled via weighted Fourier frames. We establish a sharp sampling density which is sufficient to guarantee stable recovery, without imposing any separation condition on the sampling points. In particular for the stable NUGS recovery, we also provide sufficient sampling bandwidths in the case of one-dimensional wavelet reconstructions and show sufficient linear scaling of the sampling bandwidth and the number of wavelets.