Abstract
Given two compact disjoint subsetsE 1,E 2 of the complex plane, the third problem of Zolotarev concerns estimates for the ratio
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.
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Communicated by Doron S. Lubinsky
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Levin, A.L., Saff, E.B. Optimal ray sequences of rational functions connected with the Zolotarev problem. Constr. Approx 10, 235–273 (1994). https://doi.org/10.1007/BF01263066
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DOI: https://doi.org/10.1007/BF01263066