  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.
