Stability and McMillan degree for rational matrix interpolants
Let \(\mathbb{C}_ +\) denote the right hand plane. Fix \(n\) points \(z_ 1,\dots,z_ n\) in \(\mathbb{C}_ +\) and a positive constant \(c\). Let \(w_ 1,\dots,w_ n\) be an \(n\)-tuple of complex numbers. The classical Nevanlinna-Pick interpolation problem asks to find conditions under which there is an analytic function \(f:\mathbb{C}_ +\to D(c)\), satisfying \(f(z_ j)=w_ j\), \(1 \leq j \leq n\), where \(D(c)\) is the disk centered at zero, of radius \(c\). The necessary and sufficient condition, which goes back to the original papers of Pick and respectively Nevanlinna is that the matrix \[ \left( {c^ 2-w_ iw_ j \over z_ i+\overline z_ j} \right)_{1 \leq i, j \leq n} \] to be positive semidefinite. In that case Nevanlinna has constructed in an algorithmic way a canonical rational solution to the problem. Motivated by linear systems' theory, the authors study the above problem with symmetric data (that is \(n\) is odd and \((n-1)/2\) pairs of the interpolation data satisfy \(z_ i=-\overline z_ j\), \(c^ 2=w_ i \overline w_ j)\) and with minimial McMillan degree (maximum polynomial degree of the numerator and denumerator when the solution \(f\) is rational and expressed as an irreducible quotient of polynomials). The main result asserts that, if a solution of this problem exists and the minimal degree solution is unique, then it is stable and it coincides with the constructive Nevanlinna solution. In fact the matrix valued case (when \(w_ i\) are \(m \times n\) matrices) is analyzed and settled by well known methods at this moment. See for references the monograph: \textit{J. A. Ball}, \textit{I. Gohberg} and \textit{L. Rodman}, Interpolation of rational matrix functions (1990; Zbl 0708.15011).
scientific article; zbMATH DE number 519711
scientific article