Abstract
We study quaternionic Gabor frames based on the two-sided quaternionic windowed Fourier transform. Since classical Hilbert space based methods do not work in this case we introduce appropriated versions of translation and modulation operators. We prove Janssen’s and Walnut’s representations, as well as modified versions of the Wexler–Raz biorthogonality and Ron–Shen duality based on the concept of correlation function. We end up with a characterization of tight quaternionic Gabor frames.
Highlights
The two-sided quaternionic windowed Fourier transform is one of the most interesting cases of a quaternionic windowed Fourier transform since it is neither left- nor right-linear with respect to quaternionic constants
Studies related to Gabor frames where based on the real-valued inner product and used the rotational property of this inner product to make the Fourier transform one-sided
This setting allows us to get properties like Wexler-Rax biorthogonality, RonShen duality, and Walnut and Janssen representations. This is a major step in working with quaternionic Gabor frames
Summary
The two-sided quaternionic windowed Fourier transform is one of the most interesting cases of a quaternionic windowed Fourier transform since it is neither left- nor right-linear with respect to quaternionic constants (see [1], [2], [3]) This makes the study and, in particular, the construction of Gabor frames a challenge. While several properties like uncertainty relations and the Balian-Low theorem could be shown in this case for a deeper discussion the application of the quaternionic inner product is necessary, see for instance [6], [7] Such an application does not allow to take advantage of the Hilbert space structure of the underlying function space [8]. We are going a different way by using the duality of nonlinear modulation operators This setting allows us to get properties like Wexler-Rax biorthogonality, RonShen duality, and Walnut and Janssen representations. This is a major step in working with quaternionic Gabor frames
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