I have the following integral \begin{align*} \int_{-\infty}^\infty f(t) q(t+ax) dt \end{align*}
where a is some constant.
This integral look a lot like convolution (or correlation). My question is can it be re-write as an convolution of $ f(???)* g(???)$? What are the arguments of the two functions? For example if the integral was $\int_{-\infty}^\infty f(t) q(at+ax) dt$ then we can re-write as \begin{align*} \int_{-\infty}^\infty f(t) q(at+ax) dt= f(x)*q(ax) \end{align*}
This is what I tried I did some integral manipulation and I got this identity \begin{align*} \int f(t) q(t+ax) dt=\int f(t-ax) q(t) dt \end{align*}
A convolution integral is of the form
$$\int f(t) g(x-t) \, dt;$$
the important point is the structure of the arguments:
$$x-t \qquad \text{and} \qquad t$$
In order to write the given integral as a convolution integral, we define $\tilde{q}(t) := q(-t)$. Then
$$\int_{-\infty}^{\infty} f(t) q(t+ax) \, dt = \int_{-\infty}^{\infty} f(t) \tilde{q}(-ax-t) \, dt = (f \ast \tilde{q})(-ax).$$