A moment problem associated to rational Szegő functions

Given a distribution ψ on the interval [-π,π] and a set of basis functions ζn spanning the space S, we define the moment associated with ζ_n and ψ asM(ζ_n,ψ) =∫-π^π ζ_n(exp(iθ))dψ(θ).In the classical trigonometric moment problem one has to find a distribution ψ, given the moments M(z_n,ψ), n ∈ Z. These z_n form a basis for the space of trigonometric polynomials. Note that these polynomials have only poles at zero and infinity.In this paper, the previous problem is generalized to the case where we consider the space R of rational functions having only a finite number of distinct poles a_j and b_j, j=1,...,p. The aj are distinct finite complex numbers outside the closed unit disk and the bj are their reflections b_j=1/ã_j. This space is spanned by the basis functions ζ_0=0, ζ_n^{(j)}(z) = (z- a_j)^{-n} and η_n^{(j)}(z) = (z-b_j)^{-n} for j=1,...,p and n=1,2,... The moments for all these basis functions are given and the distribution ψ has to be found. It is shown that this generalized moment problem has a unique solution if the moments generate a positive definite inner product on the space of rational functions described above.status: publishe