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.