Construction of superoptimal approximants

Abstract

LetG be a matrix function of type m×n and suppose thatG is expressible as the sum of anH∞ function and a continuous function on the unit circle. Then it is known that there is a unique superoptimal approximant toG inH∞: that is, there is a unique analytic matrix functionQ in the open unit disc which minimizess∞(G−Q) or, in other words, which minimizes the sequence $$(s_0^\infty (G - Q),s_1^\infty (G - Q),s_2^\infty (G - Q), \ldots )$$ with respect to the lexicographic ordering, wheresj∞(F)=supx∈sj(F(z)) andsj(·) denotes thejth singular value of a matrix. We give a function-theoretic (frequency domain) algorithm for the construction of this approximant. We calculate an example to illustrate the algorithm. The construction works for rationalG, but is also valid for non-rational functions. It is based on the authors' uniqueness proof in [PY1], but contains extra ingredients required to render it practicable, notably one which obviates the need for the preliminary solution of a Nehari problem. We also establish a formula forQ in terms of the maximizing vectors of a sequence of Hankel-type operators.