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 R-n.m of all rational functions of order (n, m) are considered. Let rho(n.m) = rho(n.m)(f; E) be the distance of f in the uniform metric on E from the class R-n.m. We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation {rho(n,m(n))}(n=0)(infinity), m(n)/n --> theta is an element of (0,1] as n --> infinity. In particular, we give an upper estimate for the lim inf(n-->infinity)rho(n.m(n))(1/(n+m(n))) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators.