A time domain characterization of weak Gabor dual frames on the half real line

Abstract

The study of structured frames on function spaces has interested many mathematicians during the past more than thirty years. A traditional Gabor (wavelet) frame for $L^{2}(\Bbb R)$ is generated by a translation-and-modulation (translation-and-dilation) operator system acting on finitely many functions. Nowadays, Gabor and wavelet analysis on $L^{2}(\Bbb R)$ has seen great achievements, while Gabor analysis on $L^{2}(\Bbb R_{+})$ with $\Bbb R_{+}=[0,\,\infty)$ has not. It is because $\Bbb R$ is a group under addition while $\Bbb R_{+}$ is not. This results in $L^{2}(\Bbb R_{+})$ admitting no traditional Gabor (wavelet) system. Observe that $\Bbb R_{+}$ is a group under a new addition ``$\oplus$". In this paper, we introduce and characterize a class of Gabor systems and weak Gabor dual frames in $L^{2}(\Bbb R_{+})$ based on this new group structure. Some examples are also provided.