Regularity of certain Laurent polynomials (Q2782464)

In [\textit{O. Njåstad}, J. Math. Anal. Appl. 197, No. 1, 227-248 (1996; Zbl 0853.44004)] the following Nevanlinna parameterization characterizing the solutions of a regular indeterminate strong Hamburger moment problem was proved: There exists (for a given parameter \(x_0 \in {\mathbb R}\setminus \{0\}\)) a one-to-one correspondence between all Pick functions \(\varphi\) and all solutions \(\mu\) of the moment problem. The correspondence is given by NEWLINE\[NEWLINE\int_{-\infty}^{\infty}\frac{d\mu(t)}{z-t} = \frac{\alpha(z)\varphi(z) - \gamma(z)}{\beta(z)\varphi(z) - \delta(z)},NEWLINE\]NEWLINE where \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) are obtained as limits of certain Laurent polynomials \(\alpha_n\), \(\beta_n\), \(\gamma_n\), \(\delta_n\). The proof used the fact that \(\beta_n\) are regular for infinitely many indices \(n\). It was proved for all except possibly a countable number of values of \(x_0\). NEWLINENEWLINENEWLINEIn the present note the author shows that \(\beta_n\) are regular for infinitely many indices \(n\) and thus the Nevanlinna parameterization holds for all parameters \(x_0\) except possibly one.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00021].