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I have a function $f:{\mathbb R}\rightarrow {\mathbb R}_+$ which has a unique maximum at $x=0$. $f$ can be symmetric or asymmetric. I am interested on the mollified-f function

$$\tilde{f}(x)=\int_{-\epsilon}^{\epsilon}\varphi(y)f(x-y)dy,\,\,\, \epsilon>0$$

I wonder if there are general results on the Mollifier $\varphi$, such that $\tilde{f}$ also has a unique maximum (not necessarily at $0$).

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    $\begingroup$ Questions like this have been treated in the concept of "scale space methods" in mathematical image processing about 15 or 20 years ago where one was interested in the question: How many local minima and maxima survive after convolution and can new local minima or maxima be created? Unfortunately I forgot about the exact references but one result was along the lines: If you take a scale kernel $\phi_t(x) = 1/t\phi(x/t)$ and require that the number of local maxima and minima of $\phi_t\ast f$ is decaying with $t$, and some more hypotheses, then $\phi$ is a Gaussian… $\endgroup$
    – Dirk
    Commented Nov 3, 2014 at 15:11
  • $\begingroup$ @Dirk Many thanks for the helpful pointer. $\endgroup$
    – Mole
    Commented Nov 3, 2014 at 15:37
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    $\begingroup$ It is easy to see that any condition on the mollifier is not sufficient. You need some conditions on $f$ itself. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 3, 2014 at 19:18

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Here is my comment extended into an answer. In scale space theory one considers a family of mollifiers $\phi(x,t)$ where $t$ is a positive scale parameter. Under a number of axioms (see scale-space axioms), one of which is "non-creation of local extrema of $\phi(\cdot,t)\ast f$ if $t$ increases" and another is the semi-group property $\phi(\cdot,t)\ast\phi(\cdot,s) = \phi(\cdot,s+t)$, one can conclude that $\phi(x,t)$ has to be a Gaussian with variance $\sqrt{t}$. If one discards some axioms one obtains more possible functions. References are the classical "The structure of images" by Koenderink from 1984 or a paper by Weickert et al. on an older appearance of this concept (1959 by Iijima): "Linear scale space have first been proposed in Japan" from 1999.

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