A continuous one-parameter set of binary operators on 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.