I found that
$$ \int f(t + x) g(x) dx = f \ast g^{-} (t), $$
where $g^{-}(x) = g(-x)$.
My question is that following integral
$$ \int f(t + \lambda x) g(x) dx, ~~~ \lambda \in (0, 1) $$
can be expressed in terms of convolution.
Thanks in advance.
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It's a cool idea. The change in domain between the product convolution and sum convolution is a potential problem, if $t=0$ it might be like the Mellin convolution here: https://www.emis.de/journals/BMMSS/pdf/v25n2/v25n2p1.pdf – Benedict W. J. Irwin Mar 25 '20 at 14:08