A New Theory for Fourier Analysis. Part III. Basic Analysis on the Dual

The dual to the space of analytic test functions is introduced along with operators generalizing translation, scaling, differentiation, and the Fourier-Laplace transform. Baste aspects and formulas concerning Fourier analysis on this dual is developed. It is shown that all elements of this dual space can be represented using Integration on Cn. It is also shown that this dual contains all exponentially integrable functions. Results derived include formulas for the Fourier transforms of arbitrary exponential functions and step functions.