Abstract
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.
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The authors of the paper would like to express grateful thanks to the anonymous reviewers for their professional comments and suggestions, which are greatly helpful to improve the quality of the paper.
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Communicated by Helmuth Robert Malonek
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This work was supported by the National Natural Science Foundation of China (Grant Nos. 11671414 and 12071229).
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Li, J., He, J. Some Results for the Two-Sided Quaternionic Gabor Fourier Transform and Quaternionic Gabor Frame Operator. Adv. Appl. Clifford Algebras 31, 1 (2021). https://doi.org/10.1007/s00006-020-01101-8
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DOI: https://doi.org/10.1007/s00006-020-01101-8