Approximation of sgn(x) on Two Symmetric Intervals by Rational Functions with Fixed Poles

Recently, A. Eremenko and P. Yuditskii found explicit solutions of the best polynomial approximation problems of sgn(x) over two intervals in terms of conformal mappings onto special comb domains. We give analogous solutions for the best approximation problems of sgn(x) over two symmetric intervals by odd rational functions with fixed poles. Here, the existence of the related conformal mapping is proved using convexity of the comb domains along the imaginary axis.