Characterizations of Irregular Multigenerator Gabor Frame on Periodic Subsets of ℝ

Abstract

We consider the multigenerator system $\{E_{m{b}_l}{T}_{n{a}_l}{\phi }_l,m$,$n\in \mathbb{Z},l=0,\dots ,r-1\}$ for ${\phi }_0,\dots ,{\phi }_{r-1}\in L^2(𝕊)$ and $a_0,b_0,\dots ,a_{r-1}$, $b_{r-1}>0$, where the parameters $b_0,\dots ,b_{r-1}>0$ are not necessary the same. With the help of frame theory, we provide some sufficient or necessary conditions for the system to be a frame for $L^2(𝕊)$. Moreover, we present some characterizations for this system to be a Parseval frame.