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A classical fact (due to Polya ?) is that if $P\in{\mathbb R}[X]$ has only real roots (one says that $P$ is hyperbolic), and $a$ is a real number, then the roots of $$L_aP(X):=\frac12(P(X+ia) +P(X-ia))$$ are real, too. One says that $L_a$ is hyperbolicity preserving. Of course, we may iterate: the operators $L_a$ span an abelian semi-group $S$. Notice that $L\in S$ is uni-triangular in the basis $\{1,X,X^2,\ldots\}$. Thus we may consider the closure $\bar S$ unambiguously, by considering the closure on each finite dimensional subspaces ${\mathbb R}_n[X]$.

My question is how to describe more explicitly $\bar S$. Since $L_a$ is a convolution by $\frac12(\delta_{z=-ia}+\delta_{z=ia})$, $\bar S$ consists only in convolutions by probabilities. These probabilities, supported by the imaginary axis, are symmetric with respect to the origin. After a rotation by $90$ degrees, we obtain that $\bar S$ is isomorphic to some set $\Sigma$ of symmetric probabilities over $\mathbb R$. It seems clear that normalized Gaussians, for instance, belong to $\Sigma$.

Is there a simple, explicit characterization of $\Sigma$ ?

And the following question is interesting too:

Let $\mu$ be a symmetric probability over $\mathbb R$, such that the convolution $L_\mu P:=P\star\mu(i\cdot)$ is hyperbolicity preserving. Does it belong to $\Sigma$ ?

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  • $\begingroup$ If you look at this in the Fourier domain, the Fourier transforms of these measures take the form of products $\prod_{j=1}^n cos(a_j u)$, and their pointwise limits. That includes Gaussians, finite products of such cosine factors, certain convergent infinite products of these cosines with $a_j\to 0$, and products of gaussians and such finite/infinite products. I think that covers it. I doubt you can hope to get a much simpler description of this class of functions. $\endgroup$
    – Dan Romik
    Commented Aug 31, 2022 at 23:41
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    $\begingroup$ By the way, Polya’s result is more general and also covers maps of the form $P \mapsto e^{i\theta} P(z+i a) + e^{-i \theta} P(z- i a)$ (with $a$ and $\theta$ both real), so if you allow for such things to be included then the question potentially becomes more interesting. $\endgroup$
    – Dan Romik
    Commented Aug 31, 2022 at 23:45
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    $\begingroup$ By the way 2: there’s yet more general hyperbolicity preserving maps of this flavor, see for example the one in the lemma on page 4 of the paper A local Riemann hypothesis I by Bump et al. So I do believe there are versions of your question that are quite interesting and are essentially research-level open problems. See also this PhD thesis on a related topic. $\endgroup$
    – Dan Romik
    Commented Aug 31, 2022 at 23:58

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