Analytic approximation of rational matrix functions

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 function...

On Approximation to Analytic Functions by Rational Functions

Let f(z) be analytic in the interior of a rectifiable Jordan curve C and continuous in the corresponding closed region C. The relation between continuity properties of f(z) on C and degree of approximarion to f(z) by polynomials irn(z) in z of respective degrees n, n = 1, 2, • • • , has been extensively studied. In the present paper we study the relation between continuity properties of f(z) on...

Analytic Approximation of Matrix Functions and Dual Extremal Functions

We study the question of the existence of a dual extremal function for a bounded matrix function on the unit circle in connection with the problem of approximation by analytic matrix functions. We characterize the class of matrix functions, for which a dual extremal function exists in terms of the existence of a maximizing vector of the corresponding Hankel operator and in terms of certain spec...

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class Rn,m of all rational functions of order (n,m) are considered. Let ρn,m = ρn,m( f ; E) be the distance of f in the uniform metric on E from the class Rn,m . We obtain results characterizing the rate of convergence to zero of the sequen...

Approximation by Rational Functions