Vtyjkyuk

The Mellin convolution of two functions, when it exists, is of the form
$$
(f ast_M g)(t)
= int_0^infty
fleft( fracttau right)
g(tau)
fracmathrmdtautau
$$
and has the property that
$$
mathscrM(f ast_M g)(s)
= mathscrMf cdot mathscrMg
$$
where $mathscrM$ is the Mellin transform and $cdot$ is pointwise multiplication.

Sometimes, however, it can be useful to evaluate convolution integrals which are identical to the one given above, but integrating from $-infty$ to $infty$ rather than from 0 to $infty$. In other words
$$
(f ast_widehatM g)(t)
= int_-infty^infty
fleft( fracttau right)
g(tau)
fracmathrmdtautau
$$
Does there exist a “modified Mellin transform” which has the similar property that
$$
widehat,mathscrM(f ast_widehatM g)(s)
= widehat,mathscrMf cdot widehat,mathscrMg
$$
with this “bilateral” Mellin convolution?

To note, I am aware that many authors have defined a bilateral Mellin transform according to different conventions (often as a pair of unilateral Mellin transforms). So I am not asking for an “ad hoc” definition of a bilateral Mellin transform. Rather, what I am asking is, is it possible to somehow work “backward” from the above bilateral convolution definition to obtain a modified Mellin transform, which has the property that the pointwise product in this modified Mellin domain yields the above “bilateral multiplicative Mellin convolution” in the time domain?