Periodic Non-uniqueness Sets for Shift-invariant Spaces and Parity-Based Obstructions to the Frame Property for Gabor Systems

The goal of this note is twofold. First, we provide explicit examples of periodic (though not necessarily lattice) sets that give rise to Gabor systems failing to form frames. Our constructions depend only on the parity of the window function $g$. Second, for a wide range of finite-dimensional function spaces $V$ we show that $V$ contains a function $g$ such that a lattice of high density fails to generate a Gabor frame. In particular, we prove that the Gr\"ochenig-Lyubarskii theorem is sharp in the finite-dimensional space of polynomials with Gaussian weight. More precisely, for every $N\in\mathbb{N}$ and every $\alpha,\beta>0$ satisfying $\alpha\beta=\frac{1}{N+1}$, we give an explicit algorithm for finding an even or odd polynomial $p$ of degree at most $N$ such that $\mathcal{G}(p(x)e^{-\pi x^2}, \alpha\mathbb{Z} \times \beta\mathbb{Z})$ does not form a frame. The proofs are constructive, elementary, and based on linear algebra.