ABSTRACT
This chapter presents the analytic theory underlying the rational approximation problem in Hankel norm. The problem of approximating a given causal function by a rational function of reduced degree was solved by Adamjan, Arov, and Krein in 1971, using the Hankel norm criterion. The Hankel norm criterion is valued since the approximation criterion is physically relevant and solutions may be obtained analytically. The Hankel singular values of a given function then admit an attractive interpretation: they give precisely the "distance" between a given function and some rational approximant. The chapter reviews the singular value decomposition to underscore the relevance of singular values in low rank matrix approximation problems. It gives some physical insight to the Hankel norm, and presents inequalities relating the Hankel norm to the L2 and L∞ norms. The chapter is devoted to Nehari's theorem, from which the Hankel norm approximation follows as a natural generalization. It explains the analytical construct of the optimal Hankel norm approximant.
