Critical points and error rank in best \(H_2\) matrix rational approximation of fixed McMillan degree
Let \(\widetilde{H}_{2,0}\) be the Hardy space of functions (defined in the complement of the unit disk) whose Fourier coefficients \((a_n)\) are real and satisfy \(a_n=0\) when \(n\geq 0\), endowed with the \(L_2\) norm. Similarly can be defined the Hardy space \(\widetilde{H}^{p\times m}_{2,0}\) of \(p\times m\) matrix-valued functions. The authors study the following best approximation problem: Given \(F\in \widetilde{H}^{p\times m}_{2,0}\) and an integer \(n \geq 1\), minimize \(\| F-H\| _2\) when \(H\) ranges over all (matrix-valued) rational functions analytic in the complement of the unit disk and having the ``McMillan degree'' at most \(n\). The topological structure of the set of best approximants is described in terms of the Douglas-Shapiro-Shields factorization of rational functions. In the case when \(F\) is a rational matrix (vanishing at infinity) of McMillan degree \(k\) and \(H\) is critical of degree \(n,k \geq n\) then one obtains that the rational matrix \(F-H\) has the rank \(\leq k-n\). In the particular case \(k=n\) it follows that \(H=F\) is the unique best approximant. A local uniqueness result is then derived for \(F\) of degree \(n\), when \(F\) is close enough to the compact subset \(K\) of rational matrices of degree \(n\). Deep technical results from different areas of mathematics are used.
matrix rational approximation in \(H_2\)
identification of linear dynamical systems
scientific article; zbMATH DE number 1143074