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Given a convolution:
\begin{equation}
\begin{split}
g(x) * h(x) &\doteq \int_{-\infty}^{\infty} g(z) h(x-z) dz
\end{split}
\end{equation}
Do ##z## and ##x## have to be independent? For example, can one set ##x=z+y## such that:
\begin{equation}
\begin{split}
\int_{-\infty}^{\infty} g(z) h(x-z) dz&=\int_{-\infty}^{\infty} g(z) h(y) dz
\end{split}
\end{equation}
\begin{equation}
\begin{split}
g(x) * h(x) &\doteq \int_{-\infty}^{\infty} g(z) h(x-z) dz
\end{split}
\end{equation}
Do ##z## and ##x## have to be independent? For example, can one set ##x=z+y## such that:
\begin{equation}
\begin{split}
\int_{-\infty}^{\infty} g(z) h(x-z) dz&=\int_{-\infty}^{\infty} g(z) h(y) dz
\end{split}
\end{equation}