Fractional convolution

David Mustard
School of Mathematics
University of New South Wales
Sydney
Australia 2052.

Abstract:

A continuous one-parameter set of binary operators on $L^2(\R )$called fractional convolution operators and which includes those of multiplication and convolution as particular cases is constructed by means of the Condon-Bargmann fractional Fourier transform. A fractional convolution theorem generalizes the standard Fourier convolution theorems and a fractional unit distribution generalizes the unit and delta distributions. Some explicit double-integral formulas for the fractional convolution between two functions are given and the induced operation between their corresponding Wigner distributions is found.