General Theory of Optimal Hankel-Norm Approximation

Abstract

In the previous chapter we have studied in some detail the AAK theorem for the special case where the given Hankel matrix Γ has finite rank. An elementary proof of the theorem under this assumption was given, and an application of the result to the problem of system reduction has also been illustrated. In this chapter, we will study the general AAK theory, see AAK [1971], where the Hankel matrix Γ is not necessarily of finite rank and may not even be compact. We will supply a proof in detail of the general theorem. In order to give an elementary exposition that requires as little knowledge of linear operator theory as possible, our presentation is rather lengthy. As will be seen in the first section below, the solvability of the problem can be easily proved because it is easy to show that any given bounded Hankel matrix Γ has a best approximation from the set of Hankel operators with any specified finite rank. We remark, however, that one of the main contributions of the AAK theory is that an explicit closed-form solution to the problem is formulated. The derivation of this formulation is much more difficult and will contribute to the major portion of this chapter.