Solvability condition for a boundary value interpolation problem of Loewner type

Boundary interpolation and rigidity for generalized Nevanlinna functions

“…Whereas the classical Nevanlinna-Pick interpolation problem (with given values at points in C + ) is always solvable if a certain Hankel matrix containing the data is nonnegative, for the boundary interpolation problem (as for the finite moment problem) the condition H k ≥ 0 is in general not sufficient for the existence of a solution; see Theorem 8.1. This theorem is a special case of [23,Theorem 1] which was proved there in the multi-point situation via reproducing kernel spaces. For the simpler case we present here we give a more direct proof.…”

“…For the problem (1.3) restricted to Nevanlinna functions n we prove an existence and uniqueness result without invertibility of H k ; see Theorem 8.1. This theorem also follows from a more general theorem in [23,Theorem 1]; while D. R. Georgijevic uses the theory of reproducing kernel Hilbert spaces, for our special case we use facts about Hankel matrices. Finally, applying the interpolation results for generalized Nevanlinna functions and the uniqueness statement for Nevanlinna functions, we prove a rigidity theorem for generalized Nevanlinna functions.…”

Pseudo-Taylor expansions and the Carathéodory–Fejér problem

“…Solvability of Problem ∂CF P(R) is best approached through a slight relaxation of the problem, in which the final interpolation condition (f (n) (x)/n! = a n ) is replaced by an inequality (see for example [10,6,2]). The reason is that solvability of the relaxed problem, not the original one, corresponds to positivity of a Hankel matrix.…”

On boundary interpolation for matrix valued Schur functions

“…In the former we also present a matrix analogue of conditions for the solvability of a boundary Nevanlinna-Pick problem that was obtained in a recent paper of Sarason [50]. Another approach to handling boundary interpolation problems in the scalar case is presented in [29].…”