I have the following expression
$$\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{+\infty}dx~ f(x-y) e^{-x^2/4 t} \tag{1},~~\forall ~y \in \mathbb{R}.$$.
I am trying to relate it with the generalized Weierstrass transform:
$$F(y)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{+\infty}dx~ f(y-x) e^{-x^2/4 t} \tag{2}=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{+\infty}dx~ f(x) e^{-(y-x)^2/4 t},~~\forall ~y \in \mathbb{R}. $$
In Eq. (1) the relation with the Weierstrass transform of Eq. (2) is direct if we consider that $f(x-y)$ is an even function, then $f(x-y)=f(y-x)$. However, I do not have information about $f(x-y)$. Then my question: Is qualitatively equivalent Eq. (1) to Eq. (2)?