Nonuniqueness of best rational Chebyshev approximations on the unit disk

AbstractIn the past it has been unknown whether complex rational best Chebyshev approximations (BAs) on the unit disk need be unique. This paper answers this and related questions by exhibiting examples in which: (a) the BA is not unique, (b) the number of distinct BAs is arbitrarily large, (c) the BA to a real analytic function f (i.e., f(z) = f(z)) among rational functions with real coefficients is not unique, and (d) the complex BAs to such a function are better than any approximation with real coefficients. Except in case (d), our constructions hold for approximation of arbitrary type (m, n) with n ⩾ 1. Finally, by the same methods we also establish the new result that if a function is approximated on a small disk about 0 of radius ε (or on an interval of length ε), then as ε → 0, the BA need not in general approach the corresponding Padé approximant in a sense considered by J. L. Walsh