On the entropy for indeterminate moment problems

For an indeterminate Hamburger moment problem we consider an infinite family of analytic densities solving the moment problem and we prove that they all have finite (Shannon) entropy. These densities are either all bounded or all unbounded. The result is illustrated by the Al-Salam–Carlitz moment problem, where all the densities in the family are bounded.

💡 Research Summary

The paper investigates Shannon entropy for probability densities that solve an indeterminate Hamburger moment problem. Using the classical Nevanlinna parametrization, the author considers the infinite family of analytic densities obtained by fixing a constant Pick function (\varphi(z)=t+i\gamma) with ((t,\gamma)\in\mathbb H). These densities have the explicit form

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