Quaternionic one-dimensional linear canonical transform

The quaternion linear canonical transform has been proved a powerful tool in signal processing and optics. There are many related research work about quaternion linear canonical transform but theories about one-dimensional quaternion linear canonical transform still needs to be studied. In the present paper, we define quaternionic linear canonical transform of integrable (and square integrable) functions on R. It is also shown that it satisfies all the respective properties like inversion formula, linearity, convolution theorem, Parseval's identity and the product theorem.