Algebraic Interpolation

Abstract

In this chapter, we use our knowledge of Hankel operators and chain scattering matrices to solve a set of constrained interpolation problems. These are problems in which one looks for an operator that meets a collection of specifications of the following type: (1) the operator takes specific “values” at specific “points” (we shall make the notion more precise) (2) it is constrained in norm, and (3) it is causal and has minimal state dimensions. We have to limit ourselves to specifications that satisfy a precise structure, but the class is large enough for interesting applications, namely time-varying equivalents of the celebrated “H _∞ optimal control” problem or control for minimal sensitivity. Algebraic interpolation is an extension of the notion of interpolation in complex function theory, and we derive algebraic equivalents for very classical interpolation problems such as the Nevanlinna-Pick, Schur, Hermite-Fejer and Nudel’man problems.