Two New Convolutions for the Fractional Fourier Transform

In this paper we introduce two novel convolutions for the fractional Fourier transforms, and prove natural algebraic properties of the corresponding multiplications such as commutativity, associativity and distributivity, which may be useful in signal processing and other types of applications. We analyze a consequent comparison with other known convolutions, and establish necessary and sufficient conditions for the solvability of associated convolution equations of both the first and second kind in {\$}{\$}L^1({\{}{\backslash}mathbb {\{}R{\}}{\}}){\$}{\$} L 1 ( R ) and {\$}{\$}L^2 ({\{}{\backslash}mathbb {\{}R{\}}{\}}){\$}{\$} L 2 ( R ) spaces. An example satisfying the sufficient and necessary condition for the solvability of the equations is given at the end of the paper.