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Let $f(x,y)=(x-y, x+y) $ and $\varphi \in S (\mathbb R^2).$ Prove that $\varphi \circ f \in S(\mathbb R^2 ),$ where $S $ is the Schwartz space and $(x,y)\in \mathbb R^2.$

I tried to apply

$ \forall \alpha, \beta \in \mathbb N^d$

$|(x^{\alpha} \ D^{\beta}( \varphi \circ f)(X)$|= $|(x^{\alpha} \ D^{\beta} \varphi \circ f(X). D^{\beta}f(X)|$= $|(x^{\alpha} \ D^{\beta} \varphi (f(X)). Df(X)|$

If $\beta >1$ then $D^{\beta}f(X)=0$.

Any hint will be appreciated.

A. T
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  • You are in the right track. What you need to do is write $x^{n} y^{m}$ as $((\frac {x+y} 2 + \frac {x-y} 2)^{n} \times ((\frac {x+y} 2 -\frac {x-y} 2)^{m}$ and expand the powers using Binomial theorem. – Kavi Rama Murthy Jan 10 '18 at 08:22
  • Can I replace$ Df(X) $ by jacobian or determinant jacobian of f. – A. T Jan 10 '18 at 08:33

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