  The Special Affine Fourier Transform or the SAFT generalizes a number of well
known unitary transformations as well as signal processing and optics related
mathematical operations. Unlike the Fourier transform, the SAFT does not work
well with the standard convolution operation.
  Recently, Q. Xiang and K. Y. Qin introduced a new convolution operation that
is more suitable for the SAFT and by which the SAFT of the convolution of two
functions is the product of their SAFTs and a phase factor. However, their
convolution structure does not work well with the inverse transform in sofar as
the inverse transform of the product of two functions is not equal to the
convolution of the transforms. In this article we introduce a new convolution
operation that works well with both the SAFT and its inverse leading to an
analogue of the convolution and product formulas for the Fourier transform.
Furthermore, we introduce a second convolution operation that leads to the
elimination of the phase factor in the convolution formula obtained by Q. Xiang
and K. Y. Qin.
