Fast decreasing rational functions
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Necessary and sufficient conditions are obtained for the existence of sequences of rational functions of the formr n(x) =p n(x)/pn(−x), withp n a polynomial of degreen, that decrease geometrically on (0, 1] in accordance with a specified rate function. The technique of proof involves minimum energy problems for Green potentials in the presence of an external field. Applications are given for the construction of rational approximations of |x| and sgn(x) on [−1, 1] having geometric rates of convergence forx ≠ 0.
KeywordsRational Function Maximum Principle Equilibrium Problem Rational Approximation Strict Inequality
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