Optimal ray sequences of rational functions connected with the Zolotarev problem

Abstract

Given two compact disjoint subsetsE ₁,E ₂ of the complex plane, the third problem of Zolotarev concerns estimates for the ratio

$$\mathop {\sup }\limits_{z \in E_1 } |r(z)|/\mathop {\inf }\limits_{z \in E_2 } |r(z)|,$$

wherer is a rational function of degreen. We consider, more generally, the infimumZ _mn of such ratios taken over the class of all rational functionsr with numerator degreem and denominator degreen. For any “ray sequence” of integers (m, n); that is,m/n→λ,m+n→∞, we show thatZ ^/1/(m+n) _mn approaches a limitL(λ) that can be described in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. For example, we prove thatL(λ)-exp(−F(τ)), where τ=λ/(λ+1) andF(τ) is a concave function on [0,1] and we give a formula forF(τ) in terms of the equilibrium measures forE ^*₁ ∪E ^*₂ and the condenser (E ^*₁ ,E ^*₂ ), whereE ^*₁ ,E ^*₂ are suitable subsets ofE ₁,E ₂. Of particular interest is the choice for λ that yields the smallest value forL(λ). In the case whenE ₁,E ₂, are real intervals, we provide for this purpose a simple algorithm for directly computingF(τ) and for the determination of near optimal rational functionsr _mn . Furthermore, we discuss applications of our results to the approximation of the characteristic function and to the generalized alternating direction iteration method for solving Sylvester's equation.