Gabor frames for functions supported on a semi-axis

Abstract

Let $g\in L^2(\mathbb{R})$ be a strictly decreasing continuous function supported on ${\mathbb{R}}_{+}$ such that for all $t>0$ we have $g(x+t)\le q(t)g(x)$ for some $q(t)<1$. We prove that the Gabor system $$\mathcal{G}(g;\hat{I}\pm ,{\hat{I}}^2):=\{g_{m,n}{\}}_{m,n\in \mathbb{Z}}=\{e^{2\ddot{I}€i{\hat{I}}^{2}mx}g(x-\hat{I}\pm n){\}}_{m,n\in \mathbb{Z}}$$ always forms a frame in $L^2(\mathbb{R})$ for all lattice parameters $\hat{I}\pm$,${\hat{I}}^2$, $\hat{I}\pm {\hat{I}}^2\le 1$.

9 pages. Theorem 1.4 was removed because its proof contained a gap