Given a separated set Γ in R and a generator (function) g. The quasi shift-invariant space VΓp(g) is the closure in Lp-norm of the Γ-translates of g. This space is called shift-invariant if Γ=Ζ.  

We will discuss two related properties: First, we address the following question: what assumptions should be imposed on the set Γ and generator g to ensure that the set of Γ-translates of g is lp-stable?

We establish a somewhat surprising connection between the stability property of Γ-shifts and the (non-)existence of simple quasicrystals supported by the sets from the weak limits of translates of Γ. Using some deep results from the rich theory of quasicrystals, for a wide class of generators we obtain new sufficient conditions for stability. 

Second, we introduce two families of generators that admit meromorphic extension to the complex plane and enjoy there a certain periodicity property. 

We give sufficient conditions for a separated set Λ to be a set of stable sampling in terms of densities of the sets Λ and Γ. We show that these conditions are sharp in some cases.

As an application, we obtain new results on the density of semi-regular lattices of Gabor frames with generators from these two families.  

This is joint work with Alexander Ulanovskii (University of Stavanger). 

https://univienna.zoom.us/j/64895816787?pwd=L0tHVnBPUkJFQVVSR3Y2QnhVRXRGZz09