Research proof index

For two $n$-times continuously differentiable functions $f$ and $g$, the $n$th iterated derivative distributes over subtraction: $(f-g)^{(n)}=f^{(n)}-g^{(n)}.$

Let $f$ be a twice differentiable complex-valued function whose first two derivatives are integrable, and let $g$ be a twice differentiable compactly supported cutoff that equals $1$ on $[-1,1]$ and vanishes outside $(-2,2)$. Then $g(x/R)f(x)$ converges to $f$ as $R\to\infty$ in the project norm $\|h\|=\int_{\mathbb R}|h(x)|\,dx+\frac{1}{4\pi^2}\int_{\mathbb R}|h''(x)|\,dx.$