I was reviewing the Fourier transform where I came across the convolution integral. If the convolution $x(t)*y(t) = \int_{-\infty}^{\infty}x(\tau)y(t-\tau)d\tau$ be defined, how the convolution of $x(at+b)$ and $y(ct+d)$ is defined? ($a,b,c,d \in \mathbb{R}$ are arbitrary real numbers).
I tried to do it like this. If $s = at+b$ then $t = \frac{s-b}{a}$ and we can say
$$x(at+b)*y(ct+d) = x(s)*y\left(\frac{c(s-b)}{a} + d\right)$$
Therefore it is enough to know how to write the convolution of $x(t)*y(pt+q)$ in terms of $x(t)*y(t)$ (where $p = \frac{c}{a}$ and $q = d - \frac{cb}{a}$) but unfortunately I don't know how to do the latter either. I would appreciate further hints or solution.
Thanks in advance.