The general rational interpolation problem and its connection with the Nevanlinna-pick interpolation and power moment problem

Abstract

The Hankel vector approach in a recent work of the author with Zhao and Zhang on the general rational interpolation problem (GRIP) allows the problem to be reduced to what amounts to a limiting case, i.e. a high order rational interpolation at only a single node (infinity). In particular, it gives rise to a new method to find the coefficient matrix of the linear fractional parametrization of interpolants to the GRIP. This approach is extended in the present paper to the Nevanlinna-Pick (NP) interpolation problem with multiple nodes in the class of Nevanlinna functions, which can be in essence considered as a GRIP with certain symmetries. The Hankel vector, suitably adapted to the present situation, is in fact a finite nonnegative sequence relative to an axis if the problem is solvable—in particular, a finite positive sequence in the indeterminate case. This development leads to an intrinsic connection between the NP problem and problems of the truncated power moment and reconstructing generators of the Hankel vector in question, and therefore opens up certain ways for the discussion and solution of the NP problem and its relatives on the basis of the theories of the rational interpolation and algebraic moment problems.