Exercise 25, § 5

We take up again the notation of the preceding exercises; in particular, we consider a sequence $\boldsymbol{\mu}= (\mu_n)_{n\in\mathbf{N}}$ of real numbers satisfying conditions (24) and such that the sesquilinear form $b_{\boldsymbol{\mu}}$ is separating.

Suppose that $\mu_0= 1$ and that Carleman's condition is satisfied, that is to say, that the series with general term $\mu_{2n}^{-1/(2n)}$ for $n\geqslant 1$ is divergent. a) One has

for every integer $n\geqslant 1$. (To establish the second inequality, one may prove and use the inequality $(n/e)^n\leqslant n$! for $n\geqslant 1.$)

b) The series with general term $a^{-1}_n$ for $n\geqslant 1$ is divergent.

c) There exists a unique measure $\nu \in \mathscr{M}_m(\mathbf{R})$ such that $\nu (x^n) =\mu_n$ for every $n\in \mathbf{N}$.