ON ESTIMATES FOR THE RATIO OF ERRORS IN BEST RATIONAL APPROXIMATION OF ANALYTIC FUNCTIONS

Let E be an arbitrary compact subset of the extended complex plane (C) over bar with nonempty interior. For a function f continuous on E and analytic in the interior of E denote by rho(n)(f; E) the least uniform deviation of f on E from the class of all rational functions of order at most n. In this paper we show that if f is not a rational function and if K is an arbitrary compact subset of the interior of E, then Pi(n)(k=0) (rho(k)(f; K)/rho(k) (f; E)), the ratio of the errors in best rational approximation, converges to zero geometrically as n -> infinity and the rate of convergence is determined by the capacity of the condenser (partial derivative E, K). In addition, we obtain results regarding meromorphic approximation and sharp estimates of the Hadamard type determinants.