The Fourier transform is simply the frequency spectrum of a signal. The Fourier Transform is optimum when dealing with boundary value problems. The Mellin Transform has special importance in scale representation of signal because it is scale invariant transform. For control systems engineering, stability of electrical networks etc., Mellin Transform is used. It is also useful for solving the Cauchy differential equation and Wave equation with the help of Matlab. When these two transforms are combined the resultant Fourier-Finite Mellin Transform may be applied in image processing, pattern recognition, speech processing, radar signal analysis etc. Some partial differential equation may be solved by using Fourier-Finite Mellin Transform. This paper discusses an extension of Fourier-Finite Mellin Transform in the distributional generalized sense. The Twelve testing function space are defined by using Gelfand-Shilove technique. In this paper the results on countable union space are also described.
