Complete interpolating sequences in Fock spaces and their applications to Gabor frames generated by a hyperbolic secant type function

One of the important questions in the theory of Hilbert spaces of analytic functions is the existence (and description) of complete interpolating sequences for a given space or, equivalently, Riesz bases of normalized reproducing kernels. It is well known that in many spaces (e.g., Hardy, Bergman, Bargmann–Fock spaces) there are no complete interpolating sequences. In 2010, A. Borichev and Yu. Lyubarskii showed that in a Fock type space with the weight $\exp(-a\log^2|z|)$ there exist complete interpolating sequences, and this weight is, in a certain sense, critical. In 2015, together with A. Dumont, K. Kellay and A. Hartmann, we found a description of complete interpolating sequences for this space.
Unexpectedly, this result turned out to be useful in the study of so-called shift-invariant spaces, i.e. subspaces in $L^2(\mathbb{R})$ generated by integer shifts of a fixed window function. Such spaces play an important role in time-frequency analysis. Reducing the problem to an equivalent one in a Fock type space allowed us to obtain a description of interpolation sequences for spaces generated by shifts of the Gaussian function and a secant type function. As an application, results on irregular Gabor frames from time-frequency shifts of a secant type function are obtained. The talk is based on joint work with Yu. S. Belov and K. Gröchenig.