On the structure of the Gram matrix for Gabor systems generated by B-splines

Abstract

We consider the Gabor system $\mathcal{G}(g,a\mathbb{Z}\times b\mathbb{Z})$ generated by a continuous, compactly supported function $g$ over the time-frequency lattice generated by the parameters $a$ and $b$. We show that, under an appropriate ordering of the Gabor elements, certain submatrices of the Gram matrix of $\mathcal{G}(g,a\mathbb{Z}\times b\mathbb{Z})$ exhibit a block-Toeplitz structure. This structural property enables us to derive spectral results for finite sub-blocks of the Gram matrix by appealing to the spectral theory of Toeplitz matrices. In particular, we apply our results to the Gram matrix of Gabor systems generated by the $N$th-order B-spline.

Figures (5)

• Figure 1: Top: The magnitude and phase of the finite Gram matrix $G_{15} \in \mathbb{C}^{115 \times 115}$ with blocks $G_{15}^{[\ell]} \in \mathbb{C}^{15 \times 15}$ Bottom: The magnitude of the matrix $T_{15}$ with Toeplitz blocks $T^{[\ell]}_{15}$ and the phase of the matrix $H_{15}$ with Hankel blocks $H_{15}^{[\ell]}$ in the decomposition \ref{['eqn:gram:toeplitz-dot-hankel']} for $a=.25, b=1.5, n=15$.
• Figure 2: The Laurent polynomials $t^{[\ell]}(x) \in W$ associated to the Toeplitz blocks $T_{\infty}^{[\ell]} \in \mathcal{B}(l^p(\mathbb{Z}))$, $\ell=0, \hdots, 4$. The spectrum of $T_{\infty}^{[\ell]}$ is the interval $[\min t^{[\ell]}(x), \max t^{[\ell]}(x)]$. Here, $a=.3, .4, .5$ and the shades of blue correspond to different values of $b\in [1.5, 1.98]$.
• Figure 3: Width of spectrum of $t^{[\ell]}(x)$ for $a=.23$, $b=1.7$ corresponding to the $N$th order B-spline. The dashed lines show the corresponding $N$th order decay proved in Lemma \ref{['lem:decay']}.
• Figure 4: Plots of $\lambda_1(G_n)$ (top left), $\min t^{[0]}(x)$ (top right), $\lambda_n(G_n)$ (bottom left), and $\max t^{[0]}(x)$ (bottom right) for $n=15$ demonstrate the results of Corollary \ref{['corr: eigs']}.
• Figure 5: Lemma \ref{['lem:circeigconv']} gives the convergence of eigenvalues of the Toeplitz block $T_n^{[\ell]}$ to those of the circulant matrix $C_n^{[\ell]}$. Here, $T_n^{[\ell]}$ correspond to the subblocks $G_n^{[\ell]}$ of the Gram matrices $G_n$ for $\mathcal{G}(s_2,\Lambda_n)$.

Theorems & Definitions (26)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.3.1
• proof
• Corollary 3.3.2
• Proposition 3.4