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Relaxed quaternionic Gabor expansions at critical density

Shifted and modulated Gaussian functions play a vital role in the representation of signals. We extend the theory into a quaternionic setting, using two exponential kernels with two complex numbers. As a final result, we show that every continuous and quaternion-valued signal $f$ in the Wiener space can be expanded into a unique $\ell^2$ series on a lattice at critical density $1$, provided one more point is added in the middle of a cell. We call that a relaxed Gabor expansion.