Analytic approximation of matrix functions in LpLp

We consider the problem of approximation of matrix functions of class LpLp on the unit circle by matrix functions analytic in the unit disk in the norm of LpLp, 2≤p<∞2≤p<∞. For an m×nm×n matrix function ΦΦ in LpLp, we consider the Hankel operator HΦ:Hq(Cn)→H−2(Cm), 1/p+1/q=1/21/p+1/q=1/2. It turns out that the space of m×nm×n matrix functions in LpLp splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If ΦΦ is respectable, then its distance to the set of analytic matrix functions is equal to the norm of HΦHΦ. For weird matrix functions, to obtain the distance formula, we consider Hankel operators defined on spaces of matrix functions. We also describe the set of pp-badly approximable matrix functions in terms of special factorizations and give a parametrization formula for all best analytic approximants in the norm of LpLp. Finally, we introduce the notion of pp-superoptimal approximation and prove the uniqueness of a pp-superoptimal approximant for rational matrix functions.