Optimal ray sequences of rational functions connected with the Zolotarev problem
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AbstractGiven two compact disjoint subsetsE1,E2 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 infimumZmn 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 thatZmn/1/(m+n) 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 forE1*∪E2* and the condenser (E1*,E2*), whereE1*,E2* are suitable subsets ofE1,E2. Of particular interest is the choice for λ that yields the smallest value forL(λ). In the case whenE1,E2, are real intervals, we provide for this purpose a simple algorithm for directly computingF(τ) and for the determination of near optimal rational functionsrmn. 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.