Let $N$ be a large integer and $I = [aN, bN]$ for some $0 < a < b < 1$. Denote by $\chi\_I(x) = 1$ if $x \in I$, $0$ otherwise. I was wondering if there exists a smooth function $w$ with the property that $w (x) = \chi\_I(x)$ if $x \in I$ or $\operatorname{dist}(x,I)>1/2$ and
$$
\int\_{\mathbb{R}} |w^{(n)}(x)| dx \leq C\_n
$$
for all $n \geq 1$, $C\_n$ is a positive constant that depends only on $n$.  

$w^{(n)}$ is the $n$th derivative of $w$.
I am not sure if such function exists, but any comments are appreciated


I have a sum of the form $\sum\_{m \in I} f(m)$ for some $f$ and was hoping to replace the sum with $\sum\_{m \in \mathbb{Z}} w(m) f(m)$.



Existence of a smooth function that approximates a characteristic function of an interval with certain property
Consider $\rho$ be a $C^\infty$ function supported in $(-1/8,1/8)$ with integral 1 and set
$
w=\chi\_I\ast \rho,
$
so that, for $n\ge 1$, we have  

$$
w^{(n)}(x)=\bigl(\chi\_I\ast \rho^{(n)}\bigr)(x)=
\bigl(\chi'\_I \ast \rho^{(n-1)}\bigr)(x)=\rho^{(n-1)}(x-aN)-
\rho^{(n-1)}(x-bN)
$$
and $\Vert w^{(n)}\Vert\_{L^1}\le 2\Vert\rho^{(n-1)}\Vert\_{L^1}.$
It is also possible to choose $\rho$ in the Gevrey class $G^s$ with any $s>1$ in such a way that
$$
\Vert\rho^{(n-1)}\Vert\_{L^1}\le C^n n^{sn},
$$
where $C$ is a "universal" constant.


