A moment problem associated to rational Szegő functions

Abstract

Leta ₁,...,a _p be distinct points in the finite complex plane ℂ, such that |a _j|>1,j=1,..., p and let$b_j = 1/\bar \alpha _j ,$ j=1,..., p. Let μ₀, μ ^(j)_π , ν ^(j)_π j=1,..., p;n=1, 2,... be given complex numbers. We consider the following moment problem.

Find a distribution ψ on [−π, π], with infinitely many points of increase, such that

$$\begin{array}{l} \int_{ - \pi }^\pi {d\psi (\theta ) = \mu _0 ,} \\ \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - a_j )^n }} = \mu _n^{(j)} ,} \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - b_j )^n }} = v_n^{(j)} ,} j = 1,...,p;n = 1,2,.... \\ \end{array}$$

It will be shown that this problem has a unique solution if the moments generate a positive-definite Hermitian inner product on the linear space of rational functions with no poles in the extended complex plane ℂ^* outside {a ₁,...,a _p,b ₁,...,b _p}.