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Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros.
I am looking for non-trivial examples of integral transformation \begin{align} g(x)= \int f(t) h(t,x) dt \end{align} such that $f$ and $g$ have the same number of zeros. Note that positions of zeros are allowed to change.

Edit: Suppose we assume the following regularity conditions on $f$

  1. $k$ times differentiable
  2. $f$ is absolutely integrable (i.e., $\int |f(x)| dx <\infty$)

Note: please feel free to add or change these assumptions. The point here is to see some meaningful examples.

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    $\begingroup$ What do you assume about functions $f$, $g$ and $h$? For most of the continuous functions $f$ the integral will not converge so $g$ and $f$ cannot have the same number of zeroes because $g$ would not even exist. You need to be more clear about the assumptions. $\endgroup$
    – Piotr Hajlasz
    Commented Jan 18, 2019 at 17:14
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    $\begingroup$ @PiotrHajlasz Ok. I will add some regularity conditions for $f$. $\endgroup$
    – Boby
    Commented Jan 18, 2019 at 17:30
  • $\begingroup$ I doubt there are non-trivial transformations that really preserve the number of zeroes, but there are many of them that do not increase this number. An example: $h(t,x) = e^{-(t-x)} \mathbb{1}_{(0,\infty)}(t - x)$. Try searching for "variation diminishing transform" and "Pólya frequency function" for more, if this answers your question at least partially. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jan 18, 2019 at 19:50
  • $\begingroup$ @MateuszKwaśnicki Thank you. This is very good direction. $\endgroup$
    – Boby
    Commented Jan 18, 2019 at 20:49
  • $\begingroup$ @MateuszKwaśnicki Also, are there integral transforms that increase the number of zeros by say at most $m$ $\endgroup$
    – Boby
    Commented Jan 18, 2019 at 20:51

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