Generalization of the Nevanlinna-Pick interpolation problem with multiplicities for bounded real matrices (Q1578294)

The classic Nevanlinna-Pick interpolation problem, consisting in finding a bounded analytic function on the right half-plane with prescribed values at certain points, has been extended in many important applications. One of them is the so-called tangential interpolation problem, which consists in finding a square matrix valued, analytic function \(S(s)\) such that \(I-S^*(s)S(s)\) is semi-positive definite and such that \(S(s_i)A_i=B_i\) for \(1\leq i\leq n\), where \(A_i\) and \(B_i\) are given matrices whose column dimensions are the same for each \(i\) but may vary with it. In the present paper, a constructive solution is given to an extension of the tangential interpolation problem, where also derivatives of \(S\) are asked to satisfy interpolation conditions. The proof applies a systematic procedure that reduces the order of the derivatives involved in the interpolation until a stage is reached where it can be dealt with as the tangential interpolation problem. The paper contains also an algorithm describing the whole process.