Existence of a smooth function that approximates a characteristic function of an interval with certain property

Question

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 $\sum_{m \in \mathbb{Z}} w(m) f(m)$.

Answer 1

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.