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Lower bound of the derivative $(f*g_\sigma)'$ at the zero-crossing point

I am stuck with the following problem. Let consider $f$ a smooth real function such that: $f$ is negative before 0, $f$ is positive after 0, we have $|f'(0)|>0$. Let $\sigma>0$ and $g_\sigma$ ...
NancyBoy's user avatar
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3 votes
0 answers
155 views

Inequality involving convolution roots

I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is: increasing strictly convex on $(-\infty,0)$ strictly concave on $(0,+\infty)$ Let $\sigma>0$ ...
NancyBoy's user avatar
  • 393
2 votes
1 answer
112 views

Uniqueness of the zero of $f-f*G_\sigma$ with $f$ convex/concave

I am struggling with the following problem. Let $f$ be a real smooth function: strictly convex on $(-\infty,0)$, strictly concave on $(0,\infty)$, strictly increasing. For $\sigma>0$, how can one ...
NancyBoy's user avatar
  • 393
3 votes
1 answer
220 views

Is there a real/functional analytic proof of Cramér–Lévy theorem?

In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...
Analyst's user avatar
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