A class of multi-window Gabor frames and their duals on half real line

A Zak transform is a effective method to investigate the theory of Gabor frames. Due to \(\mathbb R_{+}=[0,\,\infty )\) ${\mathbb{R}}_{+}=[0,\infty )$ being not a group with usual addition, \(L^{2}(\mathbb R_{+})\) $L^2({\mathbb{R}}_{+})$ admits no usual Gabor frames and a usual Zak transform is not suitable for the study of \(L^{2}(\mathbb R_{+})\) $L^2({\mathbb{R}}_{+})$ -Gabor frames. However, \(\mathbb R_{+}\) ${\mathbb{R}}_{+}$ is a group with “ \(\oplus \) $\oplus$ ” addition. This paper addresses a class of multi-window Gabor systems associated with “ \(\oplus \) $\oplus$ ” addition. Using a Zak transform associated with “ \(\oplus \) $\oplus$ ” addition, we characterize “ \(\oplus \) $\oplus$ ”-based Gabor frames (Riesz bases, orthonormal bases) and their (weak) Gabor dual frames. Also some examples are provided.