If $f\in S(\mathbb{R}^n)$ and for all $y\in\mathbb{R}^n$ then $\tau_{y}f\in S(\mathbb{R^n})$?
$\sup_{x} |x^{\alpha} \partial_{x}^{\beta} \tau_{y}f(x)|<\infty$?
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oh! yes. My mistake – eraldcoil Feb 28 '20 at 02:12
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It is immediate from $(x+y)^\alpha = \sum {c\choose \alpha} x^cy^{\alpha-c}$ – reuns Feb 28 '20 at 03:20