Spectral properties of the Gram matrix for Gabor systems generated by B-splines

We investigate the structure and spectrum of the Gram matrix corresponding to time-frequency shifts of the second-order B-spline. In particular we show that under a specific finite sampling of the time-frequency lattice, the Gram matrix has Toeplitz block structure and is per-Hermitian, making spectral asymptotics amenable to classical Szego-type limit theorems. We also study the relationship between the spectra of the finite-dimensional Gram matrices as well as their constituent blocks and the spectrum of the infinite-dimensional Gram operator. A complete characterization of the spectrum of the Toeplitz blocks within the Gram matrix is provided as well as explicit descriptions of the asymptotics of its eigenvalues and estimates on the corresponding frame bounds. Ultimately, we expect that the spectral analysis of these Gram matrices will lead to new results for the frame set of higher order B-splines.