Two-Dimensional Quaternion Fourier Transform Method in Probability Modeling

The Fourier transform plays a crucial role in statistics, applied mathematics, and engineering sciences. In this study, we give a definition of the two-dimensional quaternion Fourier transform, which is an extension of the two-dimensional Fourier transform. We present a new convolution theorem including this transformation. We study the characteristic function in the setting of quaternion algebra and obtain the essential properties. Based on this, we seek the expected value, variance, covariance, and their basic relations to the two-dimensional quaternion Fourier transform. We illustrate the results by giving examples to see how the obtained results differ from the classical case.