On the entropy for indeterminate moment problems

Abstract

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.

13 pages, 1 figure. The proof of Theorem 1.1 has been corrected and further details added. A new result about the Al-Salam--Carlitz case is included