The question is as in the title. This seems intuitively clear but I cannot see how to prove rigorously. Let me phrase more precisely:
For a Schwartz function $f(x,y)$ on $\mathbb{R}^2$, is the function $g(x,y)$ defined as $g(x,y):=f(x+y,y)$ also a Schwartz function?
The linear combination $x+y$ in the first argument seems of some concern for me.. Could anyone help me?