Let $\psi: \mathbb R \to \Bbb R$ be a given analytic function.
How can I construct a smooth (that is, infinitely differentiable) function $\Phi:\Bbb R\to \Bbb R$ such that $\psi|_D=\Phi|_D$ for some interval $D$, and $\Phi$ is bounded above and below?
E: I've been reminded in the comments that such an analytic $\Phi$ isn't possible, I'm changing my request to just infinitely differentiable.
Red is $\psi(x)=x^3$, blue is $y=1.5$, green is $y=-1.5$. I want to construct a $\Phi(x)$ that looks exactly like $\psi$ for $-1\leq x\leq 1$ and is bounded like the black part of the diagram.