Conformable fractional Fourier transformation of tempered distributions

Abstract

In this paper we introduce a new definition of fractional Fourier transformation on the space S of Schwartz test functions and study some of its properties. It turns out that this fractional Fourier transform has many properties with the conformable fractional derivative that the conventional Fourier transform has with the conventional (standard) derivative. We establish some operational formulas for the new transform, and give a left inverse for it. We use duality to define fractional Fourier transform of tempered distributions. Finally, we give two applications to ordinary and partial differential equations.