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Consider the transform (see e.g., (5.1) in this paper): \begin{equation*} \Lambda_\mu(q)(z) := \int_{\Delta_n} q(\zeta)\,\Re\left(\frac{1+\langle\zeta,z\rangle}{1-\langle\zeta,z\rangle}\right)d\mu(\zeta), \end{equation*} where $\Delta_n := \{ z \in \mathbb{C}^n \mid \sum_{i=1}^n |z_i|\le 1 \}$, $q$ has a convergent power series (in $\zeta, \bar{\zeta}$) on $\Delta_n$, and $\mu$ is a probability measure supported on $\Delta_n$.

Question. Does there exist a choice of $\mu$ and $q$ such that $\Lambda_\mu(q)(z) = z_1z_2\cdots z_n$ at least on the set $\{z : \max_{1\le i \le n}|z_i|\le 1\}$?

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Note: This question is outside my area, so feel free to assume any amount of ignorance on my part!

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  • $\begingroup$ Doesn't this follow from Remark 5.2 of that paper? All one needs to do is take a circularly symmetric $\mu$; any such $\mu$ for which $\int_{\Delta_n} |\zeta_1\cdots\zeta_n|^2 d\mu(\zeta)\neq 0$ will work (with $q(\zeta)$ a constant times $\zeta_1\cdots \zeta_n$). $\endgroup$
    – Mike Jury
    Commented Nov 30, 2015 at 19:54
  • $\begingroup$ One second thoughts, it seems that what you wrote is truly all what I need. I was overthinking the "none of the moments vanish" part. So the task reduces to constructing a circularly symmetric measure (which should also be a probability measure on $\Delta_n$?) for which $\int_{\Delta_n}|\zeta_1\cdots\zeta_n|^2d\mu(\zeta)=1/n!$ $\endgroup$
    – Suvrit
    Commented Nov 30, 2015 at 22:27
  • $\begingroup$ Seems like what I wrote will conflict with $\mu$ being a probability measure on $\Delta_n$; it seems that perhaps a choice other than what is given by Remark 5.2 may lead to "better" measures. $\endgroup$
    – Suvrit
    Commented Dec 1, 2015 at 0:15
  • $\begingroup$ If it's just a matter of finding some $q$ and $\mu$ that work, than the requirement that $\mu$ be a probability measure doesn't matter since one can pass constants back and forth between $q$ and $\mu$, so any circularly symmetric $\mu$ with $c:=\int_{\Delta_n} |\zeta_1\cdots \zeta_n|^2 d\mu(\zeta)>0$ works, for $q(\zeta) = n!c^{-1} \zeta_1\cdots \zeta_n$. Or is your concern to find an "optimal" pair $q,\mu$ (as mentioned further down in the remark)? $\endgroup$
    – Mike Jury
    Commented Dec 1, 2015 at 1:23
  • $\begingroup$ Thanks for your comments! In fact, after your comment, I tried playing with these values, and constructed something with not such a nice value of $c$, and I got thinking about trying to get "optimal" choices of $q$ and $\mu$ --- though then we enter the domain of this being an open problem, I guess... $\endgroup$
    – Suvrit
    Commented Dec 1, 2015 at 2:08

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