Riesz bases of reproducing kernels and complete interpolating sequences in shift-invariant spaces

Existence and description of Riesz bases of normalized reproducing kernels in Hilbert spaces of analytic functions is one of classical and deep problems of analysis. While most of classical spaces do not have such bases, it is known that in Fock-type spaces with very slowly growing weights there exist Riesz bases consisting of normalized reproducing kernels. In 2015, in a joint paper with A. Dumont, K. Kellay and A. Hartmann, we found a description of Riesz bases of reproducing kernels in the Fock-type space with the weight function $exp\left(-\log^2 |z|\right)$ which has an explicit geometric form. Unexpectedly, it turned out that this result has applications in time-frequency analysis. It allowed us to give a full description of complete interpolating sequences in subspaces of $L^2(\mathbb{R})$ generated by integer shifts of the Gaussian or of a secant type function and, consequently, to obtain new results about Gabor frames with a secant type window function. This, second, part of the talk is based on joint works with Yu.S. Belov and K. Gröchenig.