E. Neiman

Indefinite moment problem was considered by M. G. Krein and H. Langer in 1979. In the present paper the general indefinite moment problem is associated with an abstract interpolation problem in generalized Nevanlinna classes. To prove the equivalence of these two problems we investigate the structure of de Branges space $H(m)$ associated with a generalized Nevanlinna function $m$. A general formula for description of the set of solutions of indefinite moment problem is found. It is shown that the Kein-Langer description can be derived from this formula by a special choice of biorthogonal system of polynomials.

The abstract interpolation problem (AIP) in the Schur class was posed by V. Katznelson, A. Kheifets, and P. Yuditskii in 1987. In the present paper we consider an analog of AIP for the generalized Nevanlinna class $N_κ(L)$ in the nondegenerate case. We associate with the data set of the AIP a symmetric linear relation $\hat A$ acting in a Pontryagin space. The description of all solutions of the AIP is reduced to the problem of description of all $L$-resolvents of this symmetric linear relation $\hat A$. The latter set is parametrized by application of the indefinite version of Kreın’s representation theory for symmetric linear relations in Pontryagin spaces developed by M. G. Kreın and H. Langer in [22] and a formula for the $L$-resolvent matrix obtained by V. Derkach and M. Malamud in [11].