On Gabor frames generated by B-splines, totally positive functions, and Hermite functions

Abstract

The frame set of a window $\varphi \in L^2(\mathbb{R})$ is the subset of all lattice parameters $(\alpha ,\beta )\in {\mathbb{R}}_{+}^2$ such that $\mathcal{G}(\varphi ,\alpha ,\beta )=\{e^{2\pi i\beta m\cdot }\varphi (\cdot -\alpha k):k,m\in \mathbb{Z}\}$ forms a frame for $L^2(\mathbb{R})$. In this paper, we investigate the frame set of B-splines, totally positive functions, and Hermite functions. We derive a sufficient condition for Gabor frames using the connection between sampling theory in shift-invariant spaces and Gabor analysis. As a consequence, we obtain a new frame region belonging to the frame set of B-splines and Hermite functions. For a class of functions that includes certain totally positive functions, we prove that for any choice of lattice parameters $\alpha ,\beta >0$ with $\alpha \beta <1,$ there exists a $\gamma >0$ depending on $\alpha \beta$ such that $\mathcal{G}(\varphi (\gamma \cdot ),\alpha ,\beta )$ forms a frame for $L^2(\mathbb{R})$.

31 pages, 11 figures, 2 tables