In this paper, we first present some properties of the two-sided quaternionic Gabor Fourier transform (QGFT) on quaternion valued function space \(L^2({\mathbb {R}}^2,{\mathbb {H}})\), such as Parseval’s formula and the characterization of the range of the two-sided QGFT. Then, we give the definitions of quaternionic Wiener space and quaternionic Gabor frame operator (QGFO), which are the generalizations in the quaternionic settings. Finally, we prove Walnut’s and Janssen’s representation theorems and other boundedness results of the QGFO.

中文翻译：

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二维四元Gabor傅里叶变换和四元Gabor框架算子的一些结果

在本文中，我们首先介绍了四元数值函数空间\（L ^ 2（{\ mathbb {R}} ^ 2，{\ mathbb {H}}）上的双面四元Gabor Fourier变换（QGFT）的一些性质\），例如Parseval公式和两侧QGFT范围的表征。然后，我们给出了四元维纳空间和四元Gabor框架算子（QGFO）的定义，它们是四元环境中的概括。最后，我们证明了QGFO的Walnut和Janssen的表示定理和其他有界结果。