Chirp-Induced Non-Separable Gabor Windows on mathbb{R}^d

We construct an explicit class of non-separable Gabor windows on $L^2(\R^d)$ by applying chirp deformations to tensor-product dual pairs on separable lattices. Starting from one-dimensional dual Gabor frames, we first obtain separable higher-dimensional dual pairs by tensorization. We then transport these systems through the unitary chirp operator $U_C f(x)=e^{\pi i x^T Cx}f(x)$ and the associated phase-space shear, obtaining Gabor systems on lower block-triangular lattices of the form $ \Lambda_{A,B,C}=\{(Ak,CAk+B\ell):k,\ell\in\Z^d\}. $ The construction preserves compact support, smoothness, frame bounds, exact duality, canonical duals, and approximate-duality errors. We also quantify chirp-induced coupling through a phase-space covariance defect and show that non-separability changes the geometry of the atoms without deteriorating deterministic or finite-section statistical robustness bounds.