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

Yan Zhang, Yun-Zhang Li


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.


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