A Naimark dilation perspective of Nevanlinna-Pick interpolation (Q1885315)

Using the Naimark dilation theorem, along with the state space method from systems theory, a parametrization for the set of all solutions to a generalized positive real tangential Nevanlinna-Pick interpolation problem is obtained. The paper is organized in seven sections plus an introduction. The first section presents a review of the Naimark dilation theorem and the corresponding realization theorems for positive real and Schur class functions. The isometric extension problem, the parametrization of all its solutions, the tangential Nevanlinna-Pick problem, and conversely, the fact that the isometric extension problem is equivalent to a simple tangential Nevanlinna-Pick problem, can be found in the second section. In the third section, some proofs of the main results presented in the introduction are given. Using the central isometric extension, a special solution to the tangential Nevanlinna-Pick interpolation problem is presented, and the central interpolant satisfies a maximum principle. Section 4 is devoted to a Schur parametrization of all solutions. In the first subsection, the parametrization is explicit, and in the second one, some information on the coefficients in the linear fractional representation is given. The section contains an example which provides some further insight into the functions used here. The fifth section concerns the case when the solution of the Lyapunov equation is strictly positive. Solutions for the central interpolant and the Schur parametrization are obtained. Section 6 analyses the singular scalar case. Here the state space is finite-dimensional. It is shown that there is a unique solution to the tangential Nevanlinna-Pick interpolation problem in the scalar case when the Lyapunov equation is positive and singular. Also, it is shown that one can always solve the Levinson interpolation problem by using white noise plus sinusoides. In the Appendix, some standard facts that are used in the paper are recalled. The first subsection concerns infinite operator matrices, the second one presents some concepts from mathematical system theory related to the notion of realization, and the third one is devoted to the maximal outer spectral factor for a positive Toeplitz operator matrix.