Rational best approximations (in a Chebyshev sense) to real functions are characterized by an equioscillating approximation error. Similar results do not hold true for rational best approximations to complex functions in general. In the present work, we consider unitary rational approximations to the exponential function on the imaginary axis, which map the imaginary axis to the unit circle. In the class of unitary rational functions, best approximations are shown to exist, to be uniquely characterized by equioscillation of a phase error, and to possess a super-linear convergence rate. Furthermore, the best approximations have full degree (i.e., non-degenerate), achieve their maximum approximation error at points of equioscillation, and interpolate at intermediate points. Asymptotic properties of poles, interpolation nodes, and equioscillation points of these approximants are studied. Three algorithms, which are found very effective to compute unitary rational approximations including candidates for best approximations, are sketched briefly. Some consequences to numerical time-integration are discussed. In particular, time propagators based on unitary best approximants are unitary, symmetric and A-stable.

翻译：（在切比雪夫意义下）实函数的有理最优逼近具有等波纹逼近误差特征。对于复函数的有理最优逼近而言，类似结论通常不成立。本文考虑虚轴上指数函数的酉有理逼近，该类逼近将虚轴映射到单位圆。我们证明了在酉有理函数类中，最优逼近存在性、相位误差等波纹唯一性特征及超线性收敛速率。此外，最优逼近具有完全次数（即非退化），在等波纹点处达到最大逼近误差，并在中间点处进行插值。系统研究了这些逼近的极点、插值节点及等波纹点的渐近性质。简要勾勒了三种高效计算酉有理逼近（包括最优逼近候选值）的算法，并讨论了其对数值时间积分的若干应用。特别地，基于酉最优逼近的时间传播算子具有酉性、对称性和A-稳定性。