Let $\psi$ be a function on $\mathbb{R}$ satisfying
- $\phi(x)\geq 0$ for any $x$ and $\psi(x)=0$ when $|x|\geq 1$.
- $\int_{-1}^1\psi(x)dx=1$
- $\int_{-1}^1x\psi(x)=0$.
- $|\psi'''(x)|\leq B$ for a constant $B$ any $x$.
It is not hard to find a function in $\mathcal{C}^{\infty}$ satisfying these conditions. For instance $$\psi(x)=\begin{cases}Ce^{\frac{1}{x^2-1}}~\mbox{if $|x|<1$}\\ 0~\mbox{otherwise}\end{cases}$$ where $C$ is a normalizer.
My question is whether it is possible to find an analytical function meeting all the conditions or find a sequence of analytic functions $\{\phi_n\}_n$ approximating $\psi$ in the sense that $\sup\{|\phi_n(x)-\psi(x)|:x\in\mathbb{R}\}\rightarrow 0(n\rightarrow\infty)$. Moreoever, $|\phi_n'''(x)|\leq B'$ for some constant $B'$.
If $\psi \in C^0_c(\mathbb{R})$ then $$\psi_n(x) = \int_{-\infty}^\infty \psi(y) n e^{- \pi n^2 (x-y)^2}dy$$ is analytic because $e^{- x^2}$ is,
and it converges uniformly to $\psi$ because by uniform continuity $|\psi(x+y)-\psi(x)|\le g(y)$ where $g$ is continuous, bounded and $g(0) = 0$ so that $$|\psi(x) - \psi_n(x)| \le \int_{-\infty}^\infty |\psi(x-y)-\psi(x)| n e^{- \pi n^2 y^2}dy \le \int_{-\infty}^\infty g(y/n) e^{- \pi y^2}dy$$