I am struggling on the following problem.
The problem
Let $f$ be a real function, smooth, Lipschitz continuous and bounded. Let assume that $f$ is negative before 0 and positive after. For $\sigma>0$, we define the gaussian convolution $E$ by :
$$E(x,\sigma)\triangleq \frac{1}{\sigma\sqrt{2\pi}}\int_\mathbb{R}f(s)e^{-\frac{(s-x)^2}{2\sigma^2}}ds.$$
We know that under these conditions, the function $E(\cdot,\sigma)$ has only one zero-crossing denoted $x_\sigma$.
The problem is: Can we found a constant $K$ such that $|x_\sigma|\leq K\sigma^2$ ?
What we know
In this post, it is shown that this approximation exists for $\sigma$ tending to zero.
It is perhaps useful to note that $E$ is the solution of the heat equation problem $\mathcal{P}$:
$$\mathcal{P} : \begin{cases}\partial_\sigma E(x,\sigma) = \sigma\partial_{xx}E(x,\sigma) \\ E(x,0) = f(x)\end{cases}.$$
- One can notice that the function is equal to: $$E(x,\sigma) = \mathbb{E}(f(x+\sigma Z)), ~Z\sim\mathcal{N}(0,1).$$
Thank you very much!