Summary: It is the purpose of this short note to highlight perhaps unexpected connections between the topics described in papers by K. Gröchenig [in this collection, 251-260 (1990; following review)] and C. E. Heil and D. Walnut [SIAM Rev. 31, No. 4, 628-666 (1989; Zbl 0683.42031)] and at the same time between various papers within a series of joint publications with Gröchenig on the two topics indicated in the title: algorithms that allow one to recover a function (or tempered distribution) \(f\) from a suitable family of coefficients, which arise as integrals of \(f\) against a countable coherent family of functions (such as Heisenberg or affine frames) and the problem of reconstructing a band-limited function from a (sufficiently rich) set of irregularly taken sampling values. As is pointed out in detail in the paper, the basic observation, which may be taken as an explanation of most common results for these two settings, concerns properties of functions which arise as convolution products with nice, integrable functions on a locally compact group.
For the entire collection see [Zbl 0756.47055].