Frames for compactly supported functions with irrational density

Abstract

We find sufficient conditions on a compactly supported function $g$, $\supp g = [a,b]$ which guarantee that the Gabor system $$\mathcal{G}(g;\hat{I}\pm ,{\hat{I}}^2)=\{e^{2\ddot{I}€i{\hat{I}}^{2}mx}g(x-\hat{I}\pm n){\}}_{m,n\in \mathbb{Z}}$$ is a frame for all $α< b-a, αβ< 1, αβ\notin\Q$. These conditions are on one hand satisfied by almost all such functions, and on the other hand are explicit enough that we can give many concrete examples of the functions $g$ which give us a frame e.g. $g(x)=\exp (\frac{1}{x^4-1})\ddot{I}{‡}_{(-1,1)}(x)$.

16 pages