Many results about regular sampling, such as the famous Shannon sampling theorem and its practical version variants valid for weighted $L^p$-spaces rely in variants of Poisson's formula, i.e. on the strict lattice structure of the sampling set. Perturbation results like the famous Kadec theorem allow small perturbations of such a situation, but do not cover a variety of cases where the sampling set is less structured, although stable recovery is still possible.

Here frame theory, or better the theory of Banach frames comes in, and one can develop iterative algorithms with guaranteed convergence (at a geometric rate), valid for families of Banach spaces of functions or distributions (so far beyond the usual Hilbert space setting of $L^2(R^d)$).

Dealing with infinite sums, taking samples at ''well spread sampling points'', verifying robustness results (e.g. against jitter error) requires the use of a family of function spaces, called Wiener Amalgam Spaces, which are well suited when it comes to combine local with global considerations under circumstances much more general than those of perturbed lattices.

In this setting one also finds an appropriate notion of ``controlled point sets'' (the relatively separated ones), which in fact can be split into a finite union of (effectively) separated point sets. This situation was in fact relevant for the formualtion of the so-called Feichtinger conjecture (which has been answered positively as a consequence of the Kadison-Singer conjecture, in 2014).

Estimates and statements based mostly on the behaviour of pointwise multipliers and convolution operators on those Wiener Amalgam Spaces will be shown demonstrating the usefulness of this machinery in the context of irregular sampling (e.g. for band-limited functions or spline-type functions) as well as in the theory of irregular Gabor transforms.