A natural convolution of quaternion valued functions and its applications

We introduce a natural convolution of two suitable quaternion valued functions on R and list down its properties. Using this convolution, first we get the convolution theorem for Fourier transform on quaternion valued functions. Next, we modify the existing definition of wavelet transform on square integrable quaternion valued functions in a natural manner so that Parseval's identity is obtained without any additional conditions. Applying the Parseval's identity, we derive the inversion formula for the wavelet transform and we also prove the other properties like linearity, continuity and injectivity. Finally, we construct two Boehmian space of quaternion valued functions and extend the wavelet transform as a continuous linear injection from one Boehmian space into the other space.