Identification and Function Theory

Abstract

We survey in these notes certain constructive aspects of how to recover an analytic function in a plane domain from complete or partial knowledge of its boundary values. This we do with an eye on identification issues for linear dynamical systems, i.e. one-dimensional deconvolution schemes, and for that reason we restrict ourselves either to the unit disk or to the half-plane because these are the domains encountered in this context. To ensure the existence of boundary values, restrictions on the growth of the function must be made, resulting in a short introduction to Hardy spaces in the next section. We hasten to say that, in any case, the problem just mentioned is ill-posed in the sense of Hadamard [32], and actually a prototypical inverse problem: the Cauchy problem for the Laplace equation. We approach it as a constrained optimization issue, which is one of the classical routes when dealing with illposedness [51]. There are of course many ways of formulating such issues; those surveyed below make connection with the quantitative spectral theory of Toeplitz and Hankel operators that are deeply linked with meromorphic approximation. Standard regularization, which consists in requiring additional smoothness on the approximate solution, would allow us here to use classical interpolation theory; this is not the path we shall follow, but we warn the reader that linear interpolation schemes are usually not so extremely efficient in the present context. An excellent source on this topic and other matters related to our subject is [39].