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In this paper, we introduce the two-sided fractional quaternion Fourier transform (FrQFT) and give some properties of it. The main results of this paper are divided into three parts. Firstly we give a definition of the FrQFT. Secondly based on properties of the two-sided QFT, we study the relationship between the two-sided QFT and the two-sided FrQFT, and give some differential properties of the two-sided FrQFT and the Parseval identity. Finally, we give an example to illustrate the application of the two-sided FrQFT and its inverse transform in solving partial differential equations.

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In this paper, we define the two‐sided fractional Clifford–Fourier transform (FrCFT). Using its properties, we get some uncertainty principles of the FrCFT. Two parts are obtained. One part is a modified uncertainty principle. The uncertainty principle states a lower bound on the spreads of two specific transform domains. It is shown that only a Gaussian‐type signal minimizes the uncertainty. We also give a Heisenberg‐type uncertainty principle. The other part is a logarithmic uncertainty principle, which may be obtained from a sharp of Pitt's inequality.

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