The Quaternion Fourier Transform of Finite Measure and its Properties

The main focus of the present article is to propose the quaternion Fourier transform of finite measure, which is a slight generalization of the Fourier trans- form of finite measure in real and complex domains. Further, we investigate its essential properties, which including translation, Parseval’s formula, positive definite, partial derivative, and measure convergence. It is shown that according to the non-commutative nature of quaternion multiplications some properties of the Fourier transform of finite measure are not valid in the quaternion Fourier transform of finite measure.