Analytic Approximation of Matrix Functions in L

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