The Balian-Low Theorem for the Windowed Quaternionic Fourier Transform

In this paper we present the Balian-Low theorem for the twosided windowed quaternionic Fourier transform (WQFT), a theorem which expresses the fact that time-frequency concentrations are incompatible with non-redundancy whenever Gabor systems form orthonormal bases or frames. Since uncertainty principles are closely connected with representations of the kernel of the Fourier transform under consideration, we construct a suitable representation for the kernel of our two-sided WQFT which in turn provides suitable Gabor systems. We proceed by deriving several important properties of the WQFT, such as shift and modulation operators, a reconstruction formula, orthogonality relations and a Heisenberg uncertainty principle for the WQFT. Finally, we establish the Balian-Low theorem for Gabor orthonormal bases associated with discrete versions of the kernels of the WQFT and of the right-sided WQFT.