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

Abstract

The frame set of a window $\varphi \in L^2\left(R\right)$ is the subset of all lattice parameters $(\alpha ,\beta )\in R_{+}^2$ such that $G(\varphi ,\alpha ,\beta )=\left\{e^{2\pi i\beta m\cdot }\varphi \right(\cdot -\alpha k):k,m\in Z\}$ forms a frame for $L^2\left(R\right)$. 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 αβ such that $G\left(\varphi \right(\gamma \cdot ),\alpha ,\beta )$ forms a frame for $L^2\left(R\right)$. Our results give explicit frame bounds.