= Fourier bounds for Schwartz functions
{c}
{title2=$p_{m,n}(\widehat\phi)\leq C_{m,n}\sum_{a,b\leq N_{m,n}}p_{a,b}(\phi)$}
= Fourier bounds on the Schwartz space
{c}
{synonym}
For the angular-frequency <Fourier transform> and <Schwartz seminorms> $p_{m,n}(\phi)=\sup_x|x^m\phi^{(n)}(x)|$, every output <seminorm> is bounded by finitely many input <seminorms>. To prove this, <differentiation under the integral sign> and <integration by parts> give
$$
\xi^m\partial_\xi^n\widehat\phi=i^{-m}\mathcal F\!\left[\partial_x^m((-ix)^n\phi)\right].
$$
The <generalized Leibniz rule>, $\|\widehat g\|_\infty\leq\|g\|_1$, and $\|x^a\phi^{(b)}\|_1\leq\pi(p_{a,b}+p_{a+2,b})$ imply
$$
p_{m,n}(\widehat\phi)\leq\pi\sum_{j=0}^{\min(m,n)}\binom mj\frac{n!}{(n-j)!}
\bigl(p_{n-j,m-j}(\phi)+p_{n-j+2,m-j}(\phi)\bigr).
$$
This proves continuity of the <Fourier transform> on the <Schwartz space> and rapid decay of every transformed <derivative>.
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